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Properties

Label 1-103-103.90-r1-0-0
Degree $1$
Conductor $103$
Sign $0.488 + 0.872i$
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  + (0.932 + 0.361i)2-s + (0.982 + 0.183i)3-s + (0.739 + 0.673i)4-s + (0.273 + 0.961i)5-s + (0.850 + 0.526i)6-s + (0.445 − 0.895i)7-s + (0.445 + 0.895i)8-s + (0.932 + 0.361i)9-s + (−0.0922 + 0.995i)10-s + (−0.932 − 0.361i)11-s + (0.602 + 0.798i)12-s + (0.445 − 0.895i)13-s + (0.739 − 0.673i)14-s + (0.0922 + 0.995i)15-s + (0.0922 + 0.995i)16-s + (−0.850 + 0.526i)17-s + ⋯
L(s)  = 1  + (0.932 + 0.361i)2-s + (0.982 + 0.183i)3-s + (0.739 + 0.673i)4-s + (0.273 + 0.961i)5-s + (0.850 + 0.526i)6-s + (0.445 − 0.895i)7-s + (0.445 + 0.895i)8-s + (0.932 + 0.361i)9-s + (−0.0922 + 0.995i)10-s + (−0.932 − 0.361i)11-s + (0.602 + 0.798i)12-s + (0.445 − 0.895i)13-s + (0.739 − 0.673i)14-s + (0.0922 + 0.995i)15-s + (0.0922 + 0.995i)16-s + (−0.850 + 0.526i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $0.488 + 0.872i$
Analytic conductor: \(11.0688\)
Root analytic conductor: \(11.0688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (90, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (1:\ ),\ 0.488 + 0.872i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.741821139 + 2.193156005i\)
\(L(\frac12)\) \(\approx\) \(3.741821139 + 2.193156005i\)
\(L(1)\) \(\approx\) \(2.415025531 + 0.9322737261i\)
\(L(1)\) \(\approx\) \(2.415025531 + 0.9322737261i\)

Euler product

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

−29.40215468107066797011290364461, −28.62472107881165124862399243884, −27.57320872433585535729795303926, −25.87555414860516539655610541215, −25.026073211454972140430124891574, −24.22804755961787300866360102636, −23.397369889141989230522622326, −21.54321200801825621546753878285, −21.167954773821447696774858150, −20.23482833754332646052515102702, −19.17135408762608032184169715575, −18.04082121257150042617742580325, −16.11870387514883929698848776909, −15.300895209053241316666796635158, −14.19973730151944827131287836486, −13.108578064496900752370370946580, −12.49989357349876919583325942157, −11.069963753907893176001677890746, −9.4251028595097256591860409426, −8.53056649241422930496327079935, −6.91669605134385535170852017569, −5.29898974316014201745107253644, −4.308576968763614557953363347469, −2.58584440104751164009465177427, −1.63564710303189983593245386860, 2.23920962390575267029740136362, 3.34406656860176918089848416974, 4.50631618813258781103648488930, 6.1939969664742360179107667712, 7.46998895663047541219163482467, 8.29818764281386199360949845707, 10.36954562602760387957159947220, 11.02595832556578814788395831011, 13.223114140858714326304553711384, 13.47726516912265093671647755445, 14.90896478923168508931224849142, 15.233068607675929677868382991023, 16.78108284261021617583338595639, 18.10697194248888742287809514015, 19.47248774511792900313590747468, 20.68213799486370733032262546804, 21.27014768801408234099555284395, 22.48390703831645934175081128434, 23.50176953995320118146634102029, 24.54667308964857201565218252052, 25.69954821134529558585148962869, 26.299871454286723178966803039560, 27.18522038766431192229253371530, 29.20011176112323897632406596356, 30.22470176845100634284253236163

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