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Properties

Label 1-953-953.284-r0-0-0
Degree $1$
Conductor $953$
Sign $0.916 + 0.399i$
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  + (0.739 + 0.673i)2-s + (−0.850 − 0.526i)3-s + (0.0922 + 0.995i)4-s + (−0.602 − 0.798i)5-s + (−0.273 − 0.961i)6-s + (0.932 − 0.361i)7-s + (−0.602 + 0.798i)8-s + (0.445 + 0.895i)9-s + (0.0922 − 0.995i)10-s + (0.932 − 0.361i)11-s + (0.445 − 0.895i)12-s + (0.445 + 0.895i)13-s + (0.932 + 0.361i)14-s + (0.0922 + 0.995i)15-s + (−0.982 + 0.183i)16-s + (−0.273 − 0.961i)17-s + ⋯
L(s)  = 1  + (0.739 + 0.673i)2-s + (−0.850 − 0.526i)3-s + (0.0922 + 0.995i)4-s + (−0.602 − 0.798i)5-s + (−0.273 − 0.961i)6-s + (0.932 − 0.361i)7-s + (−0.602 + 0.798i)8-s + (0.445 + 0.895i)9-s + (0.0922 − 0.995i)10-s + (0.932 − 0.361i)11-s + (0.445 − 0.895i)12-s + (0.445 + 0.895i)13-s + (0.932 + 0.361i)14-s + (0.0922 + 0.995i)15-s + (−0.982 + 0.183i)16-s + (−0.273 − 0.961i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(953\)
Sign: $0.916 + 0.399i$
Analytic conductor: \(4.42571\)
Root analytic conductor: \(4.42571\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{953} (284, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 953,\ (0:\ ),\ 0.916 + 0.399i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.702762399 + 0.3545780470i\)
\(L(\frac12)\) \(\approx\) \(1.702762399 + 0.3545780470i\)
\(L(1)\) \(\approx\) \(1.257608427 + 0.2174346367i\)
\(L(1)\) \(\approx\) \(1.257608427 + 0.2174346367i\)

Euler product

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

−22.047052408631817861434393415346, −21.156005014195980087429136723598, −20.20394272835782702913576661707, −19.67492327508170440301321255892, −18.4402943453071520930916942713, −18.01118093506736378370034567842, −17.137061149467455296691130703114, −15.80313299646224252600787298147, −15.22043728191666556888981384995, −14.77632662774805078007572061181, −13.83037525064128472712624886252, −12.612817431266047364920056903339, −11.92690441577321735350710754206, −11.26306648697572265299948899099, −10.795909302633617681854324510762, −9.9620553000607486350920871332, −8.92872364872969311860693156617, −7.652218357486260155919318382980, −6.50957857161815786923558241737, −5.885907430992218281040332939637, −4.87916556389200774376742092038, −4.09605358659605283853878127708, −3.410692849645455510631241599453, −2.140515605412121852835379718309, −0.93409186873973829825586731933, 0.94570641875665324846680883223, 2.02884207906503494704063720707, 3.829921983330214275122645384306, 4.38795779509453827005937438407, 5.18572535919571306436966158917, 6.08328824551718536299008002016, 6.91435747224966261374319079806, 7.80813877147368683982350879614, 8.34942608131515071761544692495, 9.48130655392777899151730314530, 11.14137100568377544882063604860, 11.64654088818793627135695069170, 12.08661203093720387391274864549, 13.07618894021008389468756016840, 13.98005883125047433704931592892, 14.39461287492820066039648969638, 15.87089302261847404218707333724, 16.231837782193107760614132888910, 16.955726199866960535511838836256, 17.63421595699312066358492478091, 18.46041463330797839488555895714, 19.518359052416211704825893547569, 20.517160856762708606133622703356, 21.1431503039653841225932547281, 22.06471294277888758026005095758

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