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Properties

Label 1-1027-1027.1005-r1-0-0
Degree $1$
Conductor $1027$
Sign $0.789 + 0.613i$
Analytic cond. $110.366$
Root an. cond. $110.366$
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  + (0.632 − 0.774i)2-s + (0.845 + 0.534i)3-s + (−0.200 − 0.979i)4-s + (−0.120 − 0.992i)5-s + (0.948 − 0.316i)6-s + (−0.0402 + 0.999i)7-s + (−0.885 − 0.464i)8-s + (0.428 + 0.903i)9-s + (−0.845 − 0.534i)10-s + (0.919 − 0.391i)11-s + (0.354 − 0.935i)12-s + (0.748 + 0.663i)14-s + (0.428 − 0.903i)15-s + (−0.919 + 0.391i)16-s + (0.200 + 0.979i)17-s + (0.970 + 0.239i)18-s + ⋯
L(s)  = 1  + (0.632 − 0.774i)2-s + (0.845 + 0.534i)3-s + (−0.200 − 0.979i)4-s + (−0.120 − 0.992i)5-s + (0.948 − 0.316i)6-s + (−0.0402 + 0.999i)7-s + (−0.885 − 0.464i)8-s + (0.428 + 0.903i)9-s + (−0.845 − 0.534i)10-s + (0.919 − 0.391i)11-s + (0.354 − 0.935i)12-s + (0.748 + 0.663i)14-s + (0.428 − 0.903i)15-s + (−0.919 + 0.391i)16-s + (0.200 + 0.979i)17-s + (0.970 + 0.239i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1027\)    =    \(13 \cdot 79\)
Sign: $0.789 + 0.613i$
Analytic conductor: \(110.366\)
Root analytic conductor: \(110.366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1027} (1005, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1027,\ (1:\ ),\ 0.789 + 0.613i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.908464795 + 0.9966724253i\)
\(L(\frac12)\) \(\approx\) \(2.908464795 + 0.9966724253i\)
\(L(1)\) \(\approx\) \(1.709587669 - 0.3315599282i\)
\(L(1)\) \(\approx\) \(1.709587669 - 0.3315599282i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.632 - 0.774i)T \)
3 \( 1 + (0.845 + 0.534i)T \)
5 \( 1 + (-0.120 - 0.992i)T \)
7 \( 1 + (-0.0402 + 0.999i)T \)
11 \( 1 + (0.919 - 0.391i)T \)
17 \( 1 + (0.200 + 0.979i)T \)
19 \( 1 + (-0.692 + 0.721i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.428 + 0.903i)T \)
31 \( 1 + (0.354 + 0.935i)T \)
37 \( 1 + (0.278 + 0.960i)T \)
41 \( 1 + (0.799 - 0.600i)T \)
43 \( 1 + (0.919 + 0.391i)T \)
47 \( 1 + (-0.970 + 0.239i)T \)
53 \( 1 + (-0.885 - 0.464i)T \)
59 \( 1 + (0.948 - 0.316i)T \)
61 \( 1 + (-0.692 + 0.721i)T \)
67 \( 1 + (-0.987 + 0.160i)T \)
71 \( 1 + (-0.845 + 0.534i)T \)
73 \( 1 + (0.748 + 0.663i)T \)
83 \( 1 + (0.748 - 0.663i)T \)
89 \( 1 + (0.845 + 0.534i)T \)
97 \( 1 + (-0.692 + 0.721i)T \)
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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

−21.34523789519355541741332428557, −20.58823055891172860168227737413, −19.696404955034310773086276885644, −19.09774387106242991381149695364, −17.90953331938790345491541261089, −17.56505915020624752064115569082, −16.52160171930804222410911598604, −15.48411303632612957029557487043, −14.88062200590063810801947384957, −14.144114079901544187555607893711, −13.70620644566229175186143429923, −12.93669871679909306509909786429, −11.86942115483803625749800921670, −11.17029764825778330146063425558, −9.76521294338146749672219741122, −9.15867727360371469829844422273, −7.73424025524557899711736887579, −7.53798288974269254809475194210, −6.67574502380372467207387349568, −6.07951862638352227399276115234, −4.427487743271629581557175983122, −3.859053580643250216008978256084, −2.99519586646956512146434654950, −2.04122793209641818455287945981, −0.425194458700270416791092764134, 1.29483291748212414812005268067, 2.00716321822359478588710093761, 3.11776853844623651984016752882, 3.9635012007280571328107584967, 4.63458056029584358233059073462, 5.5980712502905231545189677093, 6.39921909805765184335263030535, 8.16815079471585294413423115632, 8.72173398762139087375278904103, 9.35041156374103031847296169992, 10.2145229809679845139184251149, 11.13559230381706355741937291941, 12.2304523418169259044040233726, 12.59217950484569716062287285833, 13.48231001301263188252738661917, 14.61438075478343268299049477036, 14.722675253341332864220665664217, 15.921916715991328936128972199235, 16.42991106470267614058263756049, 17.65826533348605522439978197925, 18.94087564749973351203641116992, 19.270784891378235123517623193938, 20.040746081440297349410280348840, 20.80245915105697719014991227684, 21.37535558136322558534596548517

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