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Properties

Label 1-103-103.10-r1-0-0
Degree $1$
Conductor $103$
Sign $0.988 + 0.153i$
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  + (−0.850 + 0.526i)2-s + (0.273 + 0.961i)3-s + (0.445 − 0.895i)4-s + (−0.932 + 0.361i)5-s + (−0.739 − 0.673i)6-s + (0.0922 − 0.995i)7-s + (0.0922 + 0.995i)8-s + (−0.850 + 0.526i)9-s + (0.602 − 0.798i)10-s + (0.850 − 0.526i)11-s + (0.982 + 0.183i)12-s + (0.0922 − 0.995i)13-s + (0.445 + 0.895i)14-s + (−0.602 − 0.798i)15-s + (−0.602 − 0.798i)16-s + (0.739 − 0.673i)17-s + ⋯
L(s)  = 1  + (−0.850 + 0.526i)2-s + (0.273 + 0.961i)3-s + (0.445 − 0.895i)4-s + (−0.932 + 0.361i)5-s + (−0.739 − 0.673i)6-s + (0.0922 − 0.995i)7-s + (0.0922 + 0.995i)8-s + (−0.850 + 0.526i)9-s + (0.602 − 0.798i)10-s + (0.850 − 0.526i)11-s + (0.982 + 0.183i)12-s + (0.0922 − 0.995i)13-s + (0.445 + 0.895i)14-s + (−0.602 − 0.798i)15-s + (−0.602 − 0.798i)16-s + (0.739 − 0.673i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $0.988 + 0.153i$
Analytic conductor: \(11.0688\)
Root analytic conductor: \(11.0688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (1:\ ),\ 0.988 + 0.153i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9484762076 + 0.07313368856i\)
\(L(\frac12)\) \(\approx\) \(0.9484762076 + 0.07313368856i\)
\(L(1)\) \(\approx\) \(0.7021443423 + 0.2020547953i\)
\(L(1)\) \(\approx\) \(0.7021443423 + 0.2020547953i\)

Euler product

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

−29.49640587583194426827999814389, −28.142150002329890435164102505553, −27.98150215035565204925613827094, −26.39262984521761005968884350164, −25.478360197973699789590065372734, −24.53066330166840606073869234718, −23.5642002608973697594746357152, −22.06931577878029481039883010645, −20.855733108464416373266069988745, −19.51126839809971706881279376722, −19.32619286467316261449513424492, −18.1320501045922498430255281813, −17.11864141239813091040627628299, −15.84756504540163080941300019528, −14.5796309147546020221973703946, −12.84993435811468650578796344917, −11.98744418787074141040329311905, −11.44723754190938716046776596677, −9.38590955279800451850739951748, −8.56559263645158870919134593820, −7.57788100229318626343379088098, −6.36873346349031726366501987211, −4.05798892606603363844146882838, −2.48585197983995946813945865103, −1.174698617085349806766330540363, 0.643595555536048684022346668812, 3.19021066626881584086328152598, 4.47863333277693065581238798683, 6.15742908114154480323584893304, 7.68335788042872320901142440085, 8.405405102262732011921113640677, 9.933454361100231736580827505588, 10.65286346911632577859995059011, 11.74266600374899070052724626832, 14.12363251250394892382249085294, 14.72341160680312525687007421306, 16.03662398135022550873662631742, 16.525831273665386335466396953609, 17.7987416240853111761068974247, 19.26697737226673841550400805194, 19.9404021275998064261916177698, 20.858413338375930605303561930349, 22.6379818914805615992104860346, 23.24993216031193723716137902788, 24.66908722542807819990265316591, 25.71477716941484923935835458692, 26.8579001599458135566508034437, 27.19810925508562476735150958874, 27.95217279529253720986948967779, 29.49373964620600045679795936122

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