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Properties

Label 1-103-103.14-r0-0-0
Degree $1$
Conductor $103$
Sign $-0.0945 - 0.995i$
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  + (−0.273 − 0.961i)2-s + (−0.602 + 0.798i)3-s + (−0.850 + 0.526i)4-s + (−0.982 + 0.183i)5-s + (0.932 + 0.361i)6-s + (0.739 − 0.673i)7-s + (0.739 + 0.673i)8-s + (−0.273 − 0.961i)9-s + (0.445 + 0.895i)10-s + (−0.273 − 0.961i)11-s + (0.0922 − 0.995i)12-s + (0.739 − 0.673i)13-s + (−0.850 − 0.526i)14-s + (0.445 − 0.895i)15-s + (0.445 − 0.895i)16-s + (0.932 − 0.361i)17-s + ⋯
L(s)  = 1  + (−0.273 − 0.961i)2-s + (−0.602 + 0.798i)3-s + (−0.850 + 0.526i)4-s + (−0.982 + 0.183i)5-s + (0.932 + 0.361i)6-s + (0.739 − 0.673i)7-s + (0.739 + 0.673i)8-s + (−0.273 − 0.961i)9-s + (0.445 + 0.895i)10-s + (−0.273 − 0.961i)11-s + (0.0922 − 0.995i)12-s + (0.739 − 0.673i)13-s + (−0.850 − 0.526i)14-s + (0.445 − 0.895i)15-s + (0.445 − 0.895i)16-s + (0.932 − 0.361i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0945 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0945 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $-0.0945 - 0.995i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ -0.0945 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3680853605 - 0.4047061827i\)
\(L(\frac12)\) \(\approx\) \(0.3680853605 - 0.4047061827i\)
\(L(1)\) \(\approx\) \(0.5803520530 - 0.2634950716i\)
\(L(1)\) \(\approx\) \(0.5803520530 - 0.2634950716i\)

Euler product

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

−30.438732262160413331050383444458, −28.46241283658473347400031301464, −28.11357717121096830793305629296, −27.12563233379177789094465756072, −25.68625817663424939950865845019, −24.84547628345352043493148677480, −23.69739575276430011634164036550, −23.407128498179876799283912306734, −22.23334722521314797523968460031, −20.5566262367511788379199604543, −18.84609449674328882850259338092, −18.6405517222003402067262783810, −17.31889155926958373934091799374, −16.394305338233348360705266245129, −15.262027787491668103214941930861, −14.26161085037561859558625060743, −12.7221091221317447608607807425, −11.86549676702328262260823622820, −10.47490328561496163964664819403, −8.58022947910017724121685748855, −7.887227140110030224595771011519, −6.75567891048689087875411320660, −5.48438021180972694715307021182, −4.33238651802080697586618220105, −1.57406837601252276027012563662, 0.729797320171090458090053493412, 3.28459689224164605660212581605, 4.13140272591436135023312828453, 5.44006625945478897717697289959, 7.628077823567219585299176065042, 8.69703986971049732990259739641, 10.21834562549125920550892107832, 11.15546465350001730427348694657, 11.57455602266361302445892078356, 13.13950628786335529269623028916, 14.53957575416807716361066506762, 15.86970369743994690244922064125, 16.92357246004157153775074158777, 17.98558790265640432740333019747, 19.10443253936322796367840733207, 20.333931707625679876345909689716, 21.02909703242086354366244080843, 22.10723748536868670685258416084, 23.20475520987868955582293023998, 23.78507825309527445452005548161, 26.02340154599798791584093482224, 26.8717682429577375104432639806, 27.63605268610028964243106315774, 28.14499925213479408145209998072, 29.68951112768521612645258700537

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