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Properties

Label 2-1170-65.9-c1-0-25
Degree $2$
Conductor $1170$
Sign $0.998 + 0.0471i$
Analytic cond. $9.34249$
Root an. cond. $3.05654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−2 + i)5-s + (4.33 − 2.5i)7-s + 0.999i·8-s + (−2.23 − 0.133i)10-s + (1.5 − 2.59i)11-s + (−2.59 − 2.5i)13-s + 5·14-s + (−0.5 + 0.866i)16-s + (3.46 − 2i)17-s + (−0.5 − 0.866i)19-s + (−1.86 − 1.23i)20-s + (2.59 − 1.5i)22-s + (3 − 4i)25-s + (−1 − 3.46i)26-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.894 + 0.447i)5-s + (1.63 − 0.944i)7-s + 0.353i·8-s + (−0.705 − 0.0423i)10-s + (0.452 − 0.783i)11-s + (−0.720 − 0.693i)13-s + 1.33·14-s + (−0.125 + 0.216i)16-s + (0.840 − 0.485i)17-s + (−0.114 − 0.198i)19-s + (−0.417 − 0.275i)20-s + (0.553 − 0.319i)22-s + (0.600 − 0.800i)25-s + (−0.196 − 0.679i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.998 + 0.0471i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (919, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 0.998 + 0.0471i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.395050349\)
\(L(\frac12)\) \(\approx\) \(2.395050349\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 + (2 - i)T \)
13 \( 1 + (2.59 + 2.5i)T \)
good7 \( 1 + (-4.33 + 2.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.46 + 2i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-7.79 - 4.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5 + 8.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (10.3 - 6i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 7iT - 47T^{2} \)
53 \( 1 + 3iT - 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.19 + 3i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6 - 10.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 16iT - 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 + 10iT - 83T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.66 - 5i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.959639025096914476726724586926, −8.531435673976465329915605033450, −7.86897610960937106545550181368, −7.47192082434717365117292285098, −6.53131881336942963762128828020, −5.30683172389033585176192829307, −4.60047561240050186407363785929, −3.77599288584934984354922882289, −2.76759698276698226191310154036, −0.994186858313060608139304908980, 1.43186734595603856876908669797, 2.36387994421114574851446410638, 3.84496497324274422957138265796, 4.67462734962140822993081038951, 5.13554359365179694627842582842, 6.27998069465095788168720440171, 7.51821614243163417139404999647, 8.037553039777714536114946666991, 8.953162148969376594443538973133, 9.771711023327589528813331962764

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