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Properties

Label 1-103-103.37-r1-0-0
Degree $1$
Conductor $103$
Sign $0.496 - 0.868i$
Analytic cond. $11.0688$
Root an. cond. $11.0688$
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  + (0.739 + 0.673i)2-s + (−0.932 − 0.361i)3-s + (0.0922 + 0.995i)4-s + (0.850 − 0.526i)5-s + (−0.445 − 0.895i)6-s + (−0.602 − 0.798i)7-s + (−0.602 + 0.798i)8-s + (0.739 + 0.673i)9-s + (0.982 + 0.183i)10-s + (−0.739 − 0.673i)11-s + (0.273 − 0.961i)12-s + (−0.602 − 0.798i)13-s + (0.0922 − 0.995i)14-s + (−0.982 + 0.183i)15-s + (−0.982 + 0.183i)16-s + (0.445 − 0.895i)17-s + ⋯
L(s)  = 1  + (0.739 + 0.673i)2-s + (−0.932 − 0.361i)3-s + (0.0922 + 0.995i)4-s + (0.850 − 0.526i)5-s + (−0.445 − 0.895i)6-s + (−0.602 − 0.798i)7-s + (−0.602 + 0.798i)8-s + (0.739 + 0.673i)9-s + (0.982 + 0.183i)10-s + (−0.739 − 0.673i)11-s + (0.273 − 0.961i)12-s + (−0.602 − 0.798i)13-s + (0.0922 − 0.995i)14-s + (−0.982 + 0.183i)15-s + (−0.982 + 0.183i)16-s + (0.445 − 0.895i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $0.496 - 0.868i$
Analytic conductor: \(11.0688\)
Root analytic conductor: \(11.0688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (1:\ ),\ 0.496 - 0.868i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.333952926 - 0.7740859645i\)
\(L(\frac12)\) \(\approx\) \(1.333952926 - 0.7740859645i\)
\(L(1)\) \(\approx\) \(1.167427487 - 0.04754516645i\)
\(L(1)\) \(\approx\) \(1.167427487 - 0.04754516645i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (0.739 + 0.673i)T \)
3 \( 1 + (-0.932 - 0.361i)T \)
5 \( 1 + (0.850 - 0.526i)T \)
7 \( 1 + (-0.602 - 0.798i)T \)
11 \( 1 + (-0.739 - 0.673i)T \)
13 \( 1 + (-0.602 - 0.798i)T \)
17 \( 1 + (0.445 - 0.895i)T \)
19 \( 1 + (0.932 - 0.361i)T \)
23 \( 1 + (0.739 - 0.673i)T \)
29 \( 1 + (-0.850 + 0.526i)T \)
31 \( 1 + (0.982 - 0.183i)T \)
37 \( 1 + (0.273 - 0.961i)T \)
41 \( 1 + (-0.850 - 0.526i)T \)
43 \( 1 + (0.273 + 0.961i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.932 + 0.361i)T \)
59 \( 1 + (-0.602 + 0.798i)T \)
61 \( 1 + (0.445 - 0.895i)T \)
67 \( 1 + (0.602 - 0.798i)T \)
71 \( 1 + (0.850 + 0.526i)T \)
73 \( 1 + (0.850 + 0.526i)T \)
79 \( 1 + (-0.850 + 0.526i)T \)
83 \( 1 + (-0.602 - 0.798i)T \)
89 \( 1 + (-0.0922 + 0.995i)T \)
97 \( 1 + (0.445 + 0.895i)T \)
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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

−29.47446376160235140468095187191, −28.77234363469079440918198218789, −28.27422955952886919793245495356, −26.79497082158661305895104512698, −25.52806416836073407429251184868, −24.25772656410292820393639460336, −23.11102667873781865176822252373, −22.301066282962855133569123481615, −21.58307549347925642846924914030, −20.81416571259632502590994055578, −19.06989989116613589510151537855, −18.32884315949137966594921793480, −17.0676923077626696780032919707, −15.6129552265859182724524818618, −14.78701571927289735839280729468, −13.321536754846995404483375990361, −12.35390999190894127108124254283, −11.37107725725056771725916010147, −10.01190059645165147305199152292, −9.63567310369743252331019339657, −6.83321517876833774442242690797, −5.810755067326185486822497906224, −4.94274765222361169106789175176, −3.23207336391542348007200615939, −1.78880264929438783804243831104, 0.584003817444251988330973679340, 2.914529416978122938120193426621, 4.86718930511585895075621830556, 5.585594928032195659520347936843, 6.77776235474451823225266439229, 7.835641253760088651611908062, 9.67821855741773710741647254863, 11.01124564542638396164811130462, 12.50582619337719241333483908818, 13.18127031209366629028276715401, 14.03116335070577546018731838582, 15.852972618847228202059531189682, 16.602630046494998476505038274311, 17.38759318956062460632172743594, 18.43129079314554484798943352909, 20.2636823246455287913104450003, 21.33217126848717162501192816597, 22.422306449195904938141649525481, 23.08204932107411119610888481055, 24.28875505781772937471105654871, 24.818732339525939108070614652063, 26.10652423523439777245750828342, 27.1633244483531163814321482127, 28.77132629365928981295022834742, 29.475796802321431432216576027821

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