Properties

Label 1-967-967.107-r1-0-0
Degree $1$
Conductor $967$
Sign $0.989 - 0.141i$
Analytic cond. $103.918$
Root an. cond. $103.918$
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  + (0.867 + 0.497i)2-s + (−0.165 − 0.986i)3-s + (0.505 + 0.862i)4-s + (0.209 − 0.977i)5-s + (0.347 − 0.937i)6-s + (−0.999 − 0.0260i)7-s + (0.00975 + 0.999i)8-s + (−0.945 + 0.325i)9-s + (0.668 − 0.744i)10-s + (0.710 − 0.703i)11-s + (0.767 − 0.641i)12-s + (0.964 − 0.263i)13-s + (−0.854 − 0.519i)14-s + (−0.998 − 0.0455i)15-s + (−0.488 + 0.872i)16-s + (−0.0292 + 0.999i)17-s + ⋯
L(s)  = 1  + (0.867 + 0.497i)2-s + (−0.165 − 0.986i)3-s + (0.505 + 0.862i)4-s + (0.209 − 0.977i)5-s + (0.347 − 0.937i)6-s + (−0.999 − 0.0260i)7-s + (0.00975 + 0.999i)8-s + (−0.945 + 0.325i)9-s + (0.668 − 0.744i)10-s + (0.710 − 0.703i)11-s + (0.767 − 0.641i)12-s + (0.964 − 0.263i)13-s + (−0.854 − 0.519i)14-s + (−0.998 − 0.0455i)15-s + (−0.488 + 0.872i)16-s + (−0.0292 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.989 - 0.141i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.989 - 0.141i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.515553966 - 0.2495128674i\)
\(L(\frac12)\) \(\approx\) \(3.515553966 - 0.2495128674i\)
\(L(1)\) \(\approx\) \(1.714580183 - 0.1048913128i\)
\(L(1)\) \(\approx\) \(1.714580183 - 0.1048913128i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.867 + 0.497i)T \)
3 \( 1 + (-0.165 - 0.986i)T \)
5 \( 1 + (0.209 - 0.977i)T \)
7 \( 1 + (-0.999 - 0.0260i)T \)
11 \( 1 + (0.710 - 0.703i)T \)
13 \( 1 + (0.964 - 0.263i)T \)
17 \( 1 + (-0.0292 + 0.999i)T \)
19 \( 1 + (0.791 + 0.610i)T \)
23 \( 1 + (0.371 + 0.928i)T \)
29 \( 1 + (0.107 - 0.994i)T \)
31 \( 1 + (0.571 + 0.820i)T \)
37 \( 1 + (0.0422 + 0.999i)T \)
41 \( 1 + (-0.203 + 0.979i)T \)
43 \( 1 + (0.120 - 0.992i)T \)
47 \( 1 + (-0.177 - 0.984i)T \)
53 \( 1 + (0.898 - 0.439i)T \)
59 \( 1 + (-0.628 + 0.777i)T \)
61 \( 1 + (0.746 + 0.665i)T \)
67 \( 1 + (-0.425 - 0.905i)T \)
71 \( 1 + (-0.864 - 0.502i)T \)
73 \( 1 + (0.898 + 0.439i)T \)
79 \( 1 + (-0.216 - 0.976i)T \)
83 \( 1 + (0.719 - 0.694i)T \)
89 \( 1 + (0.996 + 0.0844i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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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

−21.81836428053861829292305603626, −20.83184975990193196108372448688, −20.2559934300068676662571954794, −19.43805084954337654631421774254, −18.58603416796518056783470406323, −17.7276705350005306030729837970, −16.45924306705683618694466615840, −15.85588819817142832152882104106, −15.197529236420174013052220511246, −14.31206115251739822118021779278, −13.83141716426368804537731242636, −12.76729291566282564209575396870, −11.7689025900195816529963492252, −11.13194525198856721057103924991, −10.4025071754808066563342432975, −9.57776163488735242957147844224, −9.118866296171561599774672032118, −7.15757603068974111381963669754, −6.51196518082494538601247022493, −5.78196385745769261951042787516, −4.71443044824173678997596337389, −3.82452102907586955237016762492, −3.12867007443435583409006127754, −2.38560705532630587582267928906, −0.74109842340404314629584386191, 0.83794185066026555825641334142, 1.75029238966182566278636989822, 3.17669619162966308326882202199, 3.795013584743337986194558045989, 5.15656167854424395183290197245, 6.10850266182740336127126720851, 6.24105917735067305875018737398, 7.46907287478440887738493486466, 8.38169133204447854680231509831, 8.9253522258522056884970550089, 10.31992064224083832011863313831, 11.71247211525057993529533485188, 11.98189710643102366429116126254, 13.11114103661266793639049999969, 13.35045220246645211142562254503, 14.00051848833652462934948077983, 15.24384289190779111055744754730, 16.10259440767054251479795374915, 16.7803852773699607786158308821, 17.27285330361763235582476142786, 18.30464444658969778711113528781, 19.41778154096941849509157895960, 19.88657764754583240576349217896, 20.85459709179740486683314697487, 21.67267592313436938689249060896

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