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Properties

Label 1-103-103.97-r0-0-0
Degree $1$
Conductor $103$
Sign $0.629 + 0.776i$
Analytic cond. $0.478329$
Root an. cond. $0.478329$
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  + (0.332 − 0.943i)2-s + (0.0922 + 0.995i)3-s + (−0.779 − 0.626i)4-s + (−0.389 + 0.920i)5-s + (0.969 + 0.243i)6-s + (−0.0307 + 0.999i)7-s + (−0.850 + 0.526i)8-s + (−0.982 + 0.183i)9-s + (0.739 + 0.673i)10-s + (0.650 + 0.759i)11-s + (0.552 − 0.833i)12-s + (−0.850 − 0.526i)13-s + (0.932 + 0.361i)14-s + (−0.952 − 0.303i)15-s + (0.213 + 0.976i)16-s + (0.969 − 0.243i)17-s + ⋯
L(s)  = 1  + (0.332 − 0.943i)2-s + (0.0922 + 0.995i)3-s + (−0.779 − 0.626i)4-s + (−0.389 + 0.920i)5-s + (0.969 + 0.243i)6-s + (−0.0307 + 0.999i)7-s + (−0.850 + 0.526i)8-s + (−0.982 + 0.183i)9-s + (0.739 + 0.673i)10-s + (0.650 + 0.759i)11-s + (0.552 − 0.833i)12-s + (−0.850 − 0.526i)13-s + (0.932 + 0.361i)14-s + (−0.952 − 0.303i)15-s + (0.213 + 0.976i)16-s + (0.969 − 0.243i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $0.629 + 0.776i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ 0.629 + 0.776i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8730963148 + 0.4159858834i\)
\(L(\frac12)\) \(\approx\) \(0.8730963148 + 0.4159858834i\)
\(L(1)\) \(\approx\) \(1.017477374 + 0.1378064308i\)
\(L(1)\) \(\approx\) \(1.017477374 + 0.1378064308i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (0.332 - 0.943i)T \)
3 \( 1 + (0.0922 + 0.995i)T \)
5 \( 1 + (-0.389 + 0.920i)T \)
7 \( 1 + (-0.0307 + 0.999i)T \)
11 \( 1 + (0.650 + 0.759i)T \)
13 \( 1 + (-0.850 - 0.526i)T \)
17 \( 1 + (0.969 - 0.243i)T \)
19 \( 1 + (0.816 + 0.577i)T \)
23 \( 1 + (-0.982 - 0.183i)T \)
29 \( 1 + (0.992 - 0.122i)T \)
31 \( 1 + (0.739 - 0.673i)T \)
37 \( 1 + (0.445 + 0.895i)T \)
41 \( 1 + (-0.389 - 0.920i)T \)
43 \( 1 + (0.552 + 0.833i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.908 + 0.417i)T \)
59 \( 1 + (-0.0307 - 0.999i)T \)
61 \( 1 + (-0.273 + 0.961i)T \)
67 \( 1 + (0.881 + 0.473i)T \)
71 \( 1 + (0.992 + 0.122i)T \)
73 \( 1 + (-0.602 + 0.798i)T \)
79 \( 1 + (-0.602 - 0.798i)T \)
83 \( 1 + (0.881 - 0.473i)T \)
89 \( 1 + (0.932 + 0.361i)T \)
97 \( 1 + (0.969 + 0.243i)T \)
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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

−29.93524379183662527667223177654, −28.73046579993708472569344271193, −27.32596945838726064581493771444, −26.43373867325649194847777564256, −25.21658673925931926221856149603, −24.27285820628798376346437345213, −23.825076692722900714046727219067, −22.868966896841449136873388018786, −21.482938947911293621166259211474, −20.00055104383753699692968067088, −19.20369587995652717794504476052, −17.67625843841908147287754773934, −16.88689384905990285941118245455, −16.08286379249662286423851621029, −14.29995282129651724782914406523, −13.77445647326261183934733862172, −12.55421289853506672604851905183, −11.72582999862719481337304006611, −9.46447953108873104879153806743, −8.21585887764923852323709129151, −7.42664114632029908984362481724, −6.26619412329969300058451836420, −4.8590956326343055759485446958, −3.48217548610722211322719894112, −0.93577610924417660839375114871, 2.46123630496723667991063412472, 3.429156406495263729441760054741, 4.74467852558135729277715008894, 5.98970871951696747656434267205, 8.091247292083565309708748624946, 9.68403136889072812074580533208, 10.13081848452669248841326652897, 11.666765020321276735122431192949, 12.15550263078766215000040381412, 14.19059879923810899227904428928, 14.79645678192243351953630627269, 15.73410953718823697161429883342, 17.50284800365751618776303401449, 18.634510816801604425781290290903, 19.66017568038827138772485430677, 20.597395733424776961203211600480, 21.827205628620192011649570150756, 22.366927359360160915349128987980, 23.10438623077144259414857621818, 24.87790644644708710738100932970, 26.11626641625391390946257622722, 27.31458324284772440673800435350, 27.729068305244386061375601011124, 28.862256616192816678308510261797, 30.07594960812464990757613324056

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