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Properties

Label 1-643-643.15-r0-0-0
Degree $1$
Conductor $643$
Sign $-0.408 + 0.912i$
Analytic cond. $2.98607$
Root an. cond. $2.98607$
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  + (0.863 + 0.504i)2-s + (−0.189 − 0.981i)3-s + (0.491 + 0.870i)4-s + (0.680 + 0.732i)5-s + (0.331 − 0.943i)6-s + (−0.742 + 0.669i)7-s + (−0.0146 + 0.999i)8-s + (−0.928 + 0.372i)9-s + (0.218 + 0.975i)10-s + (−0.989 − 0.146i)11-s + (0.761 − 0.647i)12-s + (0.972 + 0.232i)13-s + (−0.978 + 0.204i)14-s + (0.590 − 0.807i)15-s + (−0.516 + 0.856i)16-s + (−0.965 − 0.261i)17-s + ⋯
L(s)  = 1  + (0.863 + 0.504i)2-s + (−0.189 − 0.981i)3-s + (0.491 + 0.870i)4-s + (0.680 + 0.732i)5-s + (0.331 − 0.943i)6-s + (−0.742 + 0.669i)7-s + (−0.0146 + 0.999i)8-s + (−0.928 + 0.372i)9-s + (0.218 + 0.975i)10-s + (−0.989 − 0.146i)11-s + (0.761 − 0.647i)12-s + (0.972 + 0.232i)13-s + (−0.978 + 0.204i)14-s + (0.590 − 0.807i)15-s + (−0.516 + 0.856i)16-s + (−0.965 − 0.261i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(643\)
Sign: $-0.408 + 0.912i$
Analytic conductor: \(2.98607\)
Root analytic conductor: \(2.98607\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{643} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 643,\ (0:\ ),\ -0.408 + 0.912i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9568835011 + 1.476554972i\)
\(L(\frac12)\) \(\approx\) \(0.9568835011 + 1.476554972i\)
\(L(1)\) \(\approx\) \(1.323662471 + 0.6004289798i\)
\(L(1)\) \(\approx\) \(1.323662471 + 0.6004289798i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad643 \( 1 \)
good2 \( 1 + (0.863 + 0.504i)T \)
3 \( 1 + (-0.189 - 0.981i)T \)
5 \( 1 + (0.680 + 0.732i)T \)
7 \( 1 + (-0.742 + 0.669i)T \)
11 \( 1 + (-0.989 - 0.146i)T \)
13 \( 1 + (0.972 + 0.232i)T \)
17 \( 1 + (-0.965 - 0.261i)T \)
19 \( 1 + (0.832 + 0.554i)T \)
23 \( 1 + (-0.948 + 0.317i)T \)
29 \( 1 + (-0.989 - 0.146i)T \)
31 \( 1 + (0.590 + 0.807i)T \)
37 \( 1 + (0.891 - 0.452i)T \)
41 \( 1 + (-0.303 - 0.952i)T \)
43 \( 1 + (0.102 + 0.994i)T \)
47 \( 1 + (-0.742 + 0.669i)T \)
53 \( 1 + (-0.999 + 0.0293i)T \)
59 \( 1 + (-0.189 + 0.981i)T \)
61 \( 1 + (0.984 + 0.175i)T \)
67 \( 1 + (0.938 - 0.345i)T \)
71 \( 1 + (-0.131 - 0.991i)T \)
73 \( 1 + (0.984 + 0.175i)T \)
79 \( 1 + (-0.131 + 0.991i)T \)
83 \( 1 + (0.590 - 0.807i)T \)
89 \( 1 + (0.863 - 0.504i)T \)
97 \( 1 + (0.160 + 0.986i)T \)
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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

−22.32047166351953196402656802371, −21.963371016596199314098263908306, −20.91958615053278036852146748120, −20.31584316211086160738482546769, −20.04326452194143958311579185371, −18.5671759436259655588112700490, −17.59417017605701397934341216020, −16.50299319874509424164531706905, −15.91200583560796826853761456082, −15.29615008430746722931003671481, −14.032654255732257730455689161098, −13.31750367313932316087116815729, −12.85782580937000211391860428578, −11.546376512969515908743159781, −10.78339499860942820459985633887, −9.950306885431267008331037505764, −9.467106679225575320991329961828, −8.20236570760731501452132024732, −6.56812951605836065914022879192, −5.833746887981233482863822958117, −4.98098786588833014101378738050, −4.17466481680992071209216312814, −3.27877762832157134213293090809, −2.17981037494977484133874401898, −0.612990327030079061936353897119, 1.89328073694808575333396213969, 2.68531933172837999626109752581, 3.51798438390195273609525982800, 5.21606831910417526566698058644, 5.99514211338853646090985420986, 6.424987857870671714402026506529, 7.408381039401922069990785194756, 8.27253099432573456300494566128, 9.416145635326345740746517959687, 10.8065279372799320648165188826, 11.533730777061096437183623608614, 12.52014883660947717312902502489, 13.3302701605913569978101640541, 13.706079763867295383927088665389, 14.63931080682574224015701689407, 15.77186281400683851727003866709, 16.2491459186444518976364425014, 17.55703467794993820865309717270, 18.1760161468166963840063155174, 18.751153418251323895884709349271, 19.92144580716922711860290809788, 20.93286136900359129663001284355, 21.78040545013474412660454225013, 22.602047811366320015355271474941, 22.97909353829188258635552231580

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