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Properties

Label 1-1012-1012.259-r0-0-0
Degree $1$
Conductor $1012$
Sign $0.326 - 0.945i$
Analytic cond. $4.69970$
Root an. cond. $4.69970$
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  + (0.254 − 0.967i)3-s + (−0.870 + 0.491i)5-s + (−0.564 + 0.825i)7-s + (−0.870 − 0.491i)9-s + (−0.610 + 0.791i)13-s + (0.254 + 0.967i)15-s + (0.466 − 0.884i)17-s + (−0.985 + 0.170i)19-s + (0.654 + 0.755i)21-s + (0.516 − 0.856i)25-s + (−0.696 + 0.717i)27-s + (0.985 + 0.170i)29-s + (0.362 − 0.931i)31-s + (0.0855 − 0.996i)35-s + (0.993 − 0.113i)37-s + ⋯
L(s)  = 1  + (0.254 − 0.967i)3-s + (−0.870 + 0.491i)5-s + (−0.564 + 0.825i)7-s + (−0.870 − 0.491i)9-s + (−0.610 + 0.791i)13-s + (0.254 + 0.967i)15-s + (0.466 − 0.884i)17-s + (−0.985 + 0.170i)19-s + (0.654 + 0.755i)21-s + (0.516 − 0.856i)25-s + (−0.696 + 0.717i)27-s + (0.985 + 0.170i)29-s + (0.362 − 0.931i)31-s + (0.0855 − 0.996i)35-s + (0.993 − 0.113i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $0.326 - 0.945i$
Analytic conductor: \(4.69970\)
Root analytic conductor: \(4.69970\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1012,\ (0:\ ),\ 0.326 - 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7564539698 - 0.5389847480i\)
\(L(\frac12)\) \(\approx\) \(0.7564539698 - 0.5389847480i\)
\(L(1)\) \(\approx\) \(0.8250656373 - 0.1853250099i\)
\(L(1)\) \(\approx\) \(0.8250656373 - 0.1853250099i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.254 - 0.967i)T \)
5 \( 1 + (-0.870 + 0.491i)T \)
7 \( 1 + (-0.564 + 0.825i)T \)
13 \( 1 + (-0.610 + 0.791i)T \)
17 \( 1 + (0.466 - 0.884i)T \)
19 \( 1 + (-0.985 + 0.170i)T \)
29 \( 1 + (0.985 + 0.170i)T \)
31 \( 1 + (0.362 - 0.931i)T \)
37 \( 1 + (0.993 - 0.113i)T \)
41 \( 1 + (-0.993 - 0.113i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (0.941 - 0.336i)T \)
59 \( 1 + (0.0285 - 0.999i)T \)
61 \( 1 + (0.998 + 0.0570i)T \)
67 \( 1 + (0.654 + 0.755i)T \)
71 \( 1 + (-0.0855 - 0.996i)T \)
73 \( 1 + (-0.897 - 0.441i)T \)
79 \( 1 + (0.610 - 0.791i)T \)
83 \( 1 + (0.198 - 0.980i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (0.198 + 0.980i)T \)
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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

−21.70304320375732851496346591930, −20.95107951428687388687936276524, −20.09571784342326612210543008743, −19.647168245648395262249391661, −19.0849194570659953193355682942, −17.51481559511089115026599773596, −16.93075583340582899213230803361, −16.26373715626659200601450535865, −15.48407257975300052335019291309, −14.898583488069116432302371461550, −14.000237593939492677607255905174, −12.94836130519573865880439637792, −12.3190235512489119212392170766, −11.21423524621808220592870844, −10.395076454340907211677297063580, −9.916266349704492999752274621511, −8.734768136828735818555624965424, −8.15715243555391889252339744074, −7.26129715414375742468914702965, −6.093531704048251118982882460786, −4.96951650939757644194781451610, −4.246833974173682503061653283640, −3.54881877321077238580014205906, −2.65161908814703945586261921268, −0.880533837340419993896991824518, 0.50348481615986285833843127488, 2.14313085793958410471797462403, 2.75197315919660335744653014496, 3.7270580649596058858506573873, 4.92016012323860296218154105671, 6.180528811358415595095139303687, 6.75064833099719512684531253747, 7.582344058259286125441858625143, 8.38537254083664809207142224435, 9.18817617586924494395383788345, 10.17290253178091229938025019045, 11.51730439204594594186747263470, 11.864488228304873502408041419250, 12.61237055156544575484669138123, 13.48033571964015116890206013417, 14.52248378994207002498751112093, 14.9316120754807738747925084546, 15.971714470744482152269211644200, 16.71757424337461059466458147973, 17.80403030344583034782201851393, 18.642260861650370656170256193554, 19.06146449766826089746496564947, 19.60059087232323794792153370749, 20.50438437327367683387509042915, 21.57648059038033699164956267821

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