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Properties

Label 1-1169-1169.76-r1-0-0
Degree $1$
Conductor $1169$
Sign $0.871 - 0.491i$
Analytic cond. $125.626$
Root an. cond. $125.626$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0189 + 0.999i)2-s + (−0.614 − 0.788i)3-s + (−0.999 − 0.0378i)4-s + (−0.489 + 0.872i)5-s + (0.800 − 0.599i)6-s + (0.0567 − 0.998i)8-s + (−0.243 + 0.969i)9-s + (−0.862 − 0.505i)10-s + (−0.169 + 0.985i)11-s + (0.584 + 0.811i)12-s + (0.700 + 0.713i)13-s + (0.988 − 0.150i)15-s + (0.997 + 0.0756i)16-s + (−0.929 − 0.369i)17-s + (−0.965 − 0.261i)18-s + (−0.206 − 0.978i)19-s + ⋯
L(s)  = 1  + (−0.0189 + 0.999i)2-s + (−0.614 − 0.788i)3-s + (−0.999 − 0.0378i)4-s + (−0.489 + 0.872i)5-s + (0.800 − 0.599i)6-s + (0.0567 − 0.998i)8-s + (−0.243 + 0.969i)9-s + (−0.862 − 0.505i)10-s + (−0.169 + 0.985i)11-s + (0.584 + 0.811i)12-s + (0.700 + 0.713i)13-s + (0.988 − 0.150i)15-s + (0.997 + 0.0756i)16-s + (−0.929 − 0.369i)17-s + (−0.965 − 0.261i)18-s + (−0.206 − 0.978i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.871 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.871 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1169\)    =    \(7 \cdot 167\)
Sign: $0.871 - 0.491i$
Analytic conductor: \(125.626\)
Root analytic conductor: \(125.626\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1169} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1169,\ (1:\ ),\ 0.871 - 0.491i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1376845891 - 0.03615153570i\)
\(L(\frac12)\) \(\approx\) \(0.1376845891 - 0.03615153570i\)
\(L(1)\) \(\approx\) \(0.4920006461 + 0.3005908507i\)
\(L(1)\) \(\approx\) \(0.4920006461 + 0.3005908507i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
167 \( 1 \)
good2 \( 1 + (-0.0189 + 0.999i)T \)
3 \( 1 + (-0.614 - 0.788i)T \)
5 \( 1 + (-0.489 + 0.872i)T \)
11 \( 1 + (-0.169 + 0.985i)T \)
13 \( 1 + (0.700 + 0.713i)T \)
17 \( 1 + (-0.929 - 0.369i)T \)
19 \( 1 + (-0.206 - 0.978i)T \)
23 \( 1 + (-0.316 + 0.948i)T \)
29 \( 1 + (-0.316 - 0.948i)T \)
31 \( 1 + (-0.421 + 0.906i)T \)
37 \( 1 + (-0.243 - 0.969i)T \)
41 \( 1 + (0.644 + 0.764i)T \)
43 \( 1 + (-0.455 + 0.890i)T \)
47 \( 1 + (0.0944 + 0.995i)T \)
53 \( 1 + (-0.752 - 0.658i)T \)
59 \( 1 + (-0.929 + 0.369i)T \)
61 \( 1 + (-0.672 + 0.739i)T \)
67 \( 1 + (0.489 + 0.872i)T \)
71 \( 1 + (-0.843 + 0.537i)T \)
73 \( 1 + (-0.997 + 0.0756i)T \)
79 \( 1 + (0.974 + 0.225i)T \)
83 \( 1 + (0.0189 + 0.999i)T \)
89 \( 1 + (-0.988 - 0.150i)T \)
97 \( 1 + (-0.421 - 0.906i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.90790643230300494156664304642, −20.514812153744207077475884434695, −19.90710333285085091911267114445, −18.77938621842022072224965100641, −18.22804678288549405022343401731, −17.120471247519663413371153022382, −16.67274403316427558893136424339, −15.79709612814000474820881582629, −15.027606963055987850918418624013, −13.929600321064337056339796695160, −13.01912145385406561366961054712, −12.3858135694503987133327635250, −11.611676599228985428759094839942, −10.79095093203184407797662428769, −10.42419417850559612652267715859, −9.20258588190645202650655400658, −8.66026186018333678698995160240, −7.9646051377596441129534714282, −6.18797404853910416036425629744, −5.48644945672880186546697274705, −4.626117877707987080933690162667, −3.833682193656587483865419315681, −3.23455842790521293658822689014, −1.70269922327180524672165320840, −0.5732672806118236412590451470, 0.05673458501009274922073073968, 1.50285391092250198527641301830, 2.69334691202537565335379353958, 4.08999620710745208529662552419, 4.7804820774441069907083257945, 5.93414685904518989531594841174, 6.62458688782195997923249946704, 7.225571170699583736088641253969, 7.79354478332177772555065644322, 8.865969879103421978781561941057, 9.8026576562905215127916302712, 10.946580347646579932979411701671, 11.49582831735793319189518795635, 12.54845295787847860394869701419, 13.3391069515755050442722432950, 13.992557760928819586173119000525, 14.87066257921199771139471514315, 15.72393143415447871628744480082, 16.18387642323506630186145315455, 17.40759880619884742949311195051, 17.85811125768064102531688209046, 18.35291335525767003069816165231, 19.30844129490931718789803162174, 19.767679601350540588076177411104, 21.31689054117260818090893720631

Graph of the $Z$-function along the critical line