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Properties

Label 1-953-953.884-r0-0-0
Degree $1$
Conductor $953$
Sign $0.975 - 0.218i$
Analytic cond. $4.42571$
Root an. cond. $4.42571$
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  + (0.445 + 0.895i)2-s + (0.739 − 0.673i)3-s + (−0.602 + 0.798i)4-s + (−0.982 + 0.183i)5-s + (0.932 + 0.361i)6-s + (−0.850 + 0.526i)7-s + (−0.982 − 0.183i)8-s + (0.0922 − 0.995i)9-s + (−0.602 − 0.798i)10-s + (−0.850 + 0.526i)11-s + (0.0922 + 0.995i)12-s + (0.0922 − 0.995i)13-s + (−0.850 − 0.526i)14-s + (−0.602 + 0.798i)15-s + (−0.273 − 0.961i)16-s + (0.932 + 0.361i)17-s + ⋯
L(s)  = 1  + (0.445 + 0.895i)2-s + (0.739 − 0.673i)3-s + (−0.602 + 0.798i)4-s + (−0.982 + 0.183i)5-s + (0.932 + 0.361i)6-s + (−0.850 + 0.526i)7-s + (−0.982 − 0.183i)8-s + (0.0922 − 0.995i)9-s + (−0.602 − 0.798i)10-s + (−0.850 + 0.526i)11-s + (0.0922 + 0.995i)12-s + (0.0922 − 0.995i)13-s + (−0.850 − 0.526i)14-s + (−0.602 + 0.798i)15-s + (−0.273 − 0.961i)16-s + (0.932 + 0.361i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(953\)
Sign: $0.975 - 0.218i$
Analytic conductor: \(4.42571\)
Root analytic conductor: \(4.42571\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{953} (884, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 953,\ (0:\ ),\ 0.975 - 0.218i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.266635036 - 0.1401333059i\)
\(L(\frac12)\) \(\approx\) \(1.266635036 - 0.1401333059i\)
\(L(1)\) \(\approx\) \(1.081806147 + 0.2458199420i\)
\(L(1)\) \(\approx\) \(1.081806147 + 0.2458199420i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad953 \( 1 \)
good2 \( 1 + (0.445 + 0.895i)T \)
3 \( 1 + (0.739 - 0.673i)T \)
5 \( 1 + (-0.982 + 0.183i)T \)
7 \( 1 + (-0.850 + 0.526i)T \)
11 \( 1 + (-0.850 + 0.526i)T \)
13 \( 1 + (0.0922 - 0.995i)T \)
17 \( 1 + (0.932 + 0.361i)T \)
19 \( 1 + (0.0922 - 0.995i)T \)
23 \( 1 + (-0.273 + 0.961i)T \)
29 \( 1 + (0.932 + 0.361i)T \)
31 \( 1 + (0.0922 - 0.995i)T \)
37 \( 1 + (0.445 - 0.895i)T \)
41 \( 1 + (-0.602 + 0.798i)T \)
43 \( 1 + (0.445 - 0.895i)T \)
47 \( 1 + (0.932 - 0.361i)T \)
53 \( 1 + (0.932 + 0.361i)T \)
59 \( 1 + (0.932 + 0.361i)T \)
61 \( 1 + (-0.850 - 0.526i)T \)
67 \( 1 + (0.0922 - 0.995i)T \)
71 \( 1 + (0.0922 - 0.995i)T \)
73 \( 1 + (-0.850 - 0.526i)T \)
79 \( 1 + (0.0922 + 0.995i)T \)
83 \( 1 + (-0.850 - 0.526i)T \)
89 \( 1 + (-0.982 - 0.183i)T \)
97 \( 1 + (-0.850 + 0.526i)T \)
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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

−21.652306008374589013837040737004, −20.79512956604519553445135367375, −20.47247004218189574406616246129, −19.460478099343987761596733391, −19.06865235135417985652324798034, −18.44425664095363197137676374165, −16.675290922591039794535956683697, −16.16494852180678318783026575407, −15.47923355148305837712506558203, −14.34612085407402960319107408335, −13.96788368458283955952317639946, −12.982640112574467213171035292482, −12.211768763312526680994242628084, −11.34672716626773469142857670185, −10.30025835554902341214662390506, −10.02801161029445918408164212902, −8.83270742766405523975304293276, −8.224210498687280506429750750, −7.09838437649764228726338621413, −5.79425591259517774245393615424, −4.67224931662947086395902041249, −4.034252816785486050695881899627, −3.26886003722024475478746260937, −2.61260519886521356827752686190, −1.06125625616865073122359134445, 0.51328608606461434446806095223, 2.58684428268036924657409408405, 3.17719699685240663878547885198, 4.00012593956204626327087327226, 5.28053230854788050448153637719, 6.138653135345019420644770330464, 7.19033040899456042844205433506, 7.64797172335296886161548440771, 8.37798212783363670802562050137, 9.23906730205069985223942196649, 10.247853172052119692165399222162, 11.769450294059126945789497837783, 12.46477251161165618797432766508, 12.99612091490737009646680559996, 13.75316774058357313406671491710, 14.86016014063402220918448427598, 15.44723249568767351502641814681, 15.72686331011774216661512837383, 16.91710432129732453328406946270, 18.04904186324456817137948568287, 18.43528326584810358489927610697, 19.42031189924820500028803063662, 20.01561426597216062167954088690, 21.016251408164067792839478883659, 21.9479310355274819147515177880

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