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Properties

Label 1-967-967.195-r1-0-0
Degree $1$
Conductor $967$
Sign $0.949 + 0.313i$
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  + (0.997 + 0.0649i)2-s + (0.984 + 0.174i)3-s + (0.991 + 0.129i)4-s + (0.533 + 0.845i)5-s + (0.971 + 0.238i)6-s + (0.00325 − 0.999i)7-s + (0.981 + 0.193i)8-s + (0.938 + 0.344i)9-s + (0.477 + 0.878i)10-s + (−0.995 + 0.0974i)11-s + (0.953 + 0.300i)12-s + (−0.413 − 0.910i)13-s + (0.0682 − 0.997i)14-s + (0.377 + 0.926i)15-s + (0.966 + 0.257i)16-s + (0.833 − 0.552i)17-s + ⋯
L(s)  = 1  + (0.997 + 0.0649i)2-s + (0.984 + 0.174i)3-s + (0.991 + 0.129i)4-s + (0.533 + 0.845i)5-s + (0.971 + 0.238i)6-s + (0.00325 − 0.999i)7-s + (0.981 + 0.193i)8-s + (0.938 + 0.344i)9-s + (0.477 + 0.878i)10-s + (−0.995 + 0.0974i)11-s + (0.953 + 0.300i)12-s + (−0.413 − 0.910i)13-s + (0.0682 − 0.997i)14-s + (0.377 + 0.926i)15-s + (0.966 + 0.257i)16-s + (0.833 − 0.552i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.949 + 0.313i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.949 + 0.313i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(7.742155313 + 1.244014888i\)
\(L(\frac12)\) \(\approx\) \(7.742155313 + 1.244014888i\)
\(L(1)\) \(\approx\) \(3.178035904 + 0.3813670641i\)
\(L(1)\) \(\approx\) \(3.178035904 + 0.3813670641i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.997 + 0.0649i)T \)
3 \( 1 + (0.984 + 0.174i)T \)
5 \( 1 + (0.533 + 0.845i)T \)
7 \( 1 + (0.00325 - 0.999i)T \)
11 \( 1 + (-0.995 + 0.0974i)T \)
13 \( 1 + (-0.413 - 0.910i)T \)
17 \( 1 + (0.833 - 0.552i)T \)
19 \( 1 + (0.889 + 0.457i)T \)
23 \( 1 + (-0.241 - 0.970i)T \)
29 \( 1 + (0.544 + 0.838i)T \)
31 \( 1 + (-0.120 + 0.992i)T \)
37 \( 1 + (0.979 - 0.200i)T \)
41 \( 1 + (0.576 - 0.816i)T \)
43 \( 1 + (0.209 - 0.977i)T \)
47 \( 1 + (-0.818 + 0.574i)T \)
53 \( 1 + (0.746 + 0.665i)T \)
59 \( 1 + (0.483 + 0.875i)T \)
61 \( 1 + (0.995 + 0.0909i)T \)
67 \( 1 + (0.799 - 0.600i)T \)
71 \( 1 + (-0.442 + 0.896i)T \)
73 \( 1 + (0.746 - 0.665i)T \)
79 \( 1 + (-0.985 - 0.168i)T \)
83 \( 1 + (0.771 + 0.636i)T \)
89 \( 1 + (-0.919 - 0.392i)T \)
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

−21.40316124643130556709043877923, −21.019178063942339912963756787406, −20.10572797123058391923866153695, −19.376738082534895273835851151443, −18.61965391965528260200870995289, −17.614895286022115718061599178415, −16.337606181786719664435337441200, −15.87677123336822462645389616418, −14.9653054313415642769089749584, −14.32942653950107948555332870697, −13.33106146590958312273662651723, −13.06065763913937443380096692180, −12.10846861801056260853743381774, −11.447731280352223256816345897881, −9.79124085429619745032123749415, −9.60833032132407171214042920689, −8.24211976618943175182594145239, −7.74283750195743347589662797412, −6.47389920696880617494939536029, −5.573976606792025399526579860767, −4.87503065306170009071081306886, −3.86548112866626065495895673512, −2.71240112989424583302217585845, −2.177325141459974544284373018, −1.172786781786078253250391952669, 1.139965537565397399833603585734, 2.4618029223583424608528395076, 2.996932095211071952692463734295, 3.75957030761039426719947344796, 4.895644638895902091306459773021, 5.667339136537190435269306440457, 7.02342987910527771251738161004, 7.40583513536107616786111982918, 8.20432425609328554671390839841, 9.8339792838879578278484940970, 10.323354015895863329886462388683, 10.88684906259401808109391208032, 12.33941715488472376767260255166, 13.02549949319672853758179046459, 13.91519644708481058745150711429, 14.24403683979775823371065722657, 14.945742926061597700984376987831, 15.895195467317312165211804020540, 16.48509214204378649628968241202, 17.76159745475782170342851233221, 18.51150421321562931636785981843, 19.54568806178493577115147266427, 20.31787889234018089115066598012, 20.78788706299920076419914364521, 21.52599770025831838874123868003

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