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Properties

Label 1-307-307.269-r0-0-0
Degree $1$
Conductor $307$
Sign $0.992 - 0.122i$
Analytic cond. $1.42570$
Root an. cond. $1.42570$
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 + (−0.850 − 0.526i)3-s + (−0.602 − 0.798i)4-s + (−0.602 + 0.798i)5-s + (−0.850 + 0.526i)6-s + (0.932 − 0.361i)7-s + (−0.982 + 0.183i)8-s + (0.445 + 0.895i)9-s + (0.445 + 0.895i)10-s + (−0.850 + 0.526i)11-s + (0.0922 + 0.995i)12-s + (−0.273 + 0.961i)13-s + (0.0922 − 0.995i)14-s + (0.932 − 0.361i)15-s + (−0.273 + 0.961i)16-s + 17-s + ⋯
L(s)  = 1  + (0.445 − 0.895i)2-s + (−0.850 − 0.526i)3-s + (−0.602 − 0.798i)4-s + (−0.602 + 0.798i)5-s + (−0.850 + 0.526i)6-s + (0.932 − 0.361i)7-s + (−0.982 + 0.183i)8-s + (0.445 + 0.895i)9-s + (0.445 + 0.895i)10-s + (−0.850 + 0.526i)11-s + (0.0922 + 0.995i)12-s + (−0.273 + 0.961i)13-s + (0.0922 − 0.995i)14-s + (0.932 − 0.361i)15-s + (−0.273 + 0.961i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 307 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 307 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(307\)
Sign: $0.992 - 0.122i$
Analytic conductor: \(1.42570\)
Root analytic conductor: \(1.42570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{307} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 307,\ (0:\ ),\ 0.992 - 0.122i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8563027068 - 0.05282719679i\)
\(L(\frac12)\) \(\approx\) \(0.8563027068 - 0.05282719679i\)
\(L(1)\) \(\approx\) \(0.8112761307 - 0.2870987417i\)
\(L(1)\) \(\approx\) \(0.8112761307 - 0.2870987417i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad307 \( 1 \)
good2 \( 1 + (0.445 - 0.895i)T \)
3 \( 1 + (-0.850 - 0.526i)T \)
5 \( 1 + (-0.602 + 0.798i)T \)
7 \( 1 + (0.932 - 0.361i)T \)
11 \( 1 + (-0.850 + 0.526i)T \)
13 \( 1 + (-0.273 + 0.961i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.273 + 0.961i)T \)
23 \( 1 + (0.445 + 0.895i)T \)
29 \( 1 + (0.445 - 0.895i)T \)
31 \( 1 + (-0.850 + 0.526i)T \)
37 \( 1 + (0.932 - 0.361i)T \)
41 \( 1 + (0.932 + 0.361i)T \)
43 \( 1 + (-0.982 + 0.183i)T \)
47 \( 1 + (-0.273 + 0.961i)T \)
53 \( 1 + T \)
59 \( 1 + (0.739 - 0.673i)T \)
61 \( 1 + (-0.982 + 0.183i)T \)
67 \( 1 + (0.0922 + 0.995i)T \)
71 \( 1 + (-0.982 - 0.183i)T \)
73 \( 1 + (-0.273 + 0.961i)T \)
79 \( 1 + (-0.982 - 0.183i)T \)
83 \( 1 + (0.932 + 0.361i)T \)
89 \( 1 + (0.0922 + 0.995i)T \)
97 \( 1 + (-0.273 - 0.961i)T \)
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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

−25.02467169573335118929527272443, −24.14847481376785635749568022020, −23.64861846241322088232984837129, −22.873728835627444130795999226872, −21.7495267057199200474255713295, −21.164127122149732963653724832, −20.27566118865838724273459513528, −18.54257840327498920144695897252, −17.83442292898268776606485860490, −16.8226082121999927003514217633, −16.27234282246820753448701002086, −15.271605337694144530755793645371, −14.78759575730417611521544536656, −13.19599930272672796776882880754, −12.44744380394039206971337574558, −11.58224924597860894702024049169, −10.51779285762254026916973020690, −9.034079207976743440701008564308, −8.20315632700369130272386380573, −7.29223526572996692035807455789, −5.744784140275834566172854746363, −5.15469772029753686260086586161, −4.4611536616765641583129843218, −3.11977737073231992534048605401, −0.600011925308042611069730544445, 1.3484068359823034994267828888, 2.438359695779873578914580877666, 3.95247324911388671555070616490, 4.87536992246823116635130827101, 5.92134166407558864909208861005, 7.23692203802834906768071429326, 8.01525718783027386952307341029, 9.87895869677165426570673679114, 10.66924388128764171683136828096, 11.47615343179240092365121678680, 12.04960356895957658955051768697, 13.095606302151166883600756161062, 14.21590725146351034600616136302, 14.86003032188716683788153100197, 16.19206008372746517938616161278, 17.45725008983794026340758949265, 18.29270644503736939896950308429, 18.891298359528240432877891260136, 19.75442476867231969550916102508, 21.07231735942279200302343636542, 21.58186660682724987424156622741, 22.806672547154434003801056461377, 23.47012275651838353517366686614, 23.698030932658446423998714447356, 25.013406964582783120061364331671

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