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Properties

Label 1-2349-2349.68-r0-0-0
Degree $1$
Conductor $2349$
Sign $0.740 - 0.672i$
Analytic cond. $10.9087$
Root an. cond. $10.9087$
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  + (−0.981 + 0.189i)2-s + (0.927 − 0.373i)4-s + (0.991 − 0.132i)5-s + (0.140 + 0.990i)7-s + (−0.840 + 0.542i)8-s + (−0.947 + 0.318i)10-s + (0.971 − 0.238i)11-s + (0.983 − 0.181i)13-s + (−0.326 − 0.945i)14-s + (0.721 − 0.692i)16-s + (−0.642 − 0.766i)17-s + (−0.999 − 0.0249i)19-s + (0.870 − 0.492i)20-s + (−0.908 + 0.418i)22-s + (−0.933 − 0.357i)23-s + ⋯
L(s)  = 1  + (−0.981 + 0.189i)2-s + (0.927 − 0.373i)4-s + (0.991 − 0.132i)5-s + (0.140 + 0.990i)7-s + (−0.840 + 0.542i)8-s + (−0.947 + 0.318i)10-s + (0.971 − 0.238i)11-s + (0.983 − 0.181i)13-s + (−0.326 − 0.945i)14-s + (0.721 − 0.692i)16-s + (−0.642 − 0.766i)17-s + (−0.999 − 0.0249i)19-s + (0.870 − 0.492i)20-s + (−0.908 + 0.418i)22-s + (−0.933 − 0.357i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2349\)    =    \(3^{4} \cdot 29\)
Sign: $0.740 - 0.672i$
Analytic conductor: \(10.9087\)
Root analytic conductor: \(10.9087\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2349} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2349,\ (0:\ ),\ 0.740 - 0.672i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.208737495 - 0.4667872281i\)
\(L(\frac12)\) \(\approx\) \(1.208737495 - 0.4667872281i\)
\(L(1)\) \(\approx\) \(0.8979823305 + 0.02384169911i\)
\(L(1)\) \(\approx\) \(0.8979823305 + 0.02384169911i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.981 + 0.189i)T \)
5 \( 1 + (0.991 - 0.132i)T \)
7 \( 1 + (0.140 + 0.990i)T \)
11 \( 1 + (0.971 - 0.238i)T \)
13 \( 1 + (0.983 - 0.181i)T \)
17 \( 1 + (-0.642 - 0.766i)T \)
19 \( 1 + (-0.999 - 0.0249i)T \)
23 \( 1 + (-0.933 - 0.357i)T \)
31 \( 1 + (0.214 - 0.976i)T \)
37 \( 1 + (-0.388 - 0.921i)T \)
41 \( 1 + (0.802 + 0.597i)T \)
43 \( 1 + (0.924 - 0.380i)T \)
47 \( 1 + (-0.897 - 0.441i)T \)
53 \( 1 + (-0.0747 - 0.997i)T \)
59 \( 1 + (-0.893 - 0.448i)T \)
61 \( 1 + (0.373 - 0.927i)T \)
67 \( 1 + (-0.107 - 0.994i)T \)
71 \( 1 + (0.542 - 0.840i)T \)
73 \( 1 + (-0.715 - 0.698i)T \)
79 \( 1 + (0.760 + 0.649i)T \)
83 \( 1 + (0.254 - 0.966i)T \)
89 \( 1 + (0.0995 - 0.995i)T \)
97 \( 1 + (-0.995 + 0.0912i)T \)
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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

−19.55729782105578117441427451015, −19.06888385255110645619963589116, −17.92116790302328759345405891240, −17.65473987030892839347523842176, −17.0167691761345219816882587962, −16.405710721486693485914555025805, −15.54292294289289153907658707733, −14.59688753408661838384585046010, −13.942597051896904478812100388180, −13.14784898903351152758774430759, −12.38845650398671169773627116899, −11.38320078448898223351499166805, −10.690829708525593748885983153287, −10.27681898897508356027178057162, −9.39565267516900142246054217426, −8.76205566020270516797732177116, −8.04218065902418696048513852587, −6.954709547183772707273712234542, −6.471354222541649723726101217778, −5.85867165626421068996014547655, −4.34102071669750933105220851574, −3.7402096373408495367337407092, −2.589527323252481228979382088756, −1.56931747987853634536185214986, −1.22859483252545773557096807383, 0.59360821853784721657609403109, 1.85672360652609055095589182915, 2.16554627792327730188467731339, 3.28098390612784329631615827736, 4.56221192795520815454227406427, 5.66383337844857656902013174422, 6.21581511879115172880092437526, 6.648355055109157107579576965010, 7.94315786688851238840383433537, 8.650357158658598516446662278894, 9.19150718373430582274345832196, 9.703368450594808706026919850774, 10.757013707356489859037294464082, 11.28283957085074131117609044856, 12.12877665749042859054452235402, 12.920829894415414133419318378660, 13.925870949436965194544967137, 14.55585446540014497535137809323, 15.35254624238996871526322622712, 16.10260188488046492302632624364, 16.6936969599321268855896040372, 17.59566949491625992486745670295, 17.9701994342088072604444033063, 18.63909477099281557465087294295, 19.33020478092871229081311548099

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