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Properties

Label 1-113-113.25-r0-0-0
Degree $1$
Conductor $113$
Sign $-0.552 - 0.833i$
Analytic cond. $0.524769$
Root an. cond. $0.524769$
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  + (0.222 − 0.974i)2-s + (−0.993 + 0.111i)3-s + (−0.900 − 0.433i)4-s + (0.993 + 0.111i)5-s + (−0.111 + 0.993i)6-s + (0.623 − 0.781i)7-s + (−0.623 + 0.781i)8-s + (0.974 − 0.222i)9-s + (0.330 − 0.943i)10-s + (−0.433 − 0.900i)11-s + (0.943 + 0.330i)12-s + (−0.781 − 0.623i)13-s + (−0.623 − 0.781i)14-s − 15-s + (0.623 + 0.781i)16-s + (−0.846 + 0.532i)17-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)2-s + (−0.993 + 0.111i)3-s + (−0.900 − 0.433i)4-s + (0.993 + 0.111i)5-s + (−0.111 + 0.993i)6-s + (0.623 − 0.781i)7-s + (−0.623 + 0.781i)8-s + (0.974 − 0.222i)9-s + (0.330 − 0.943i)10-s + (−0.433 − 0.900i)11-s + (0.943 + 0.330i)12-s + (−0.781 − 0.623i)13-s + (−0.623 − 0.781i)14-s − 15-s + (0.623 + 0.781i)16-s + (−0.846 + 0.532i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $-0.552 - 0.833i$
Analytic conductor: \(0.524769\)
Root analytic conductor: \(0.524769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (0:\ ),\ -0.552 - 0.833i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4115600200 - 0.7665078080i\)
\(L(\frac12)\) \(\approx\) \(0.4115600200 - 0.7665078080i\)
\(L(1)\) \(\approx\) \(0.7144820220 - 0.5687452385i\)
\(L(1)\) \(\approx\) \(0.7144820220 - 0.5687452385i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad113 \( 1 \)
good2 \( 1 + (0.222 - 0.974i)T \)
3 \( 1 + (-0.993 + 0.111i)T \)
5 \( 1 + (0.993 + 0.111i)T \)
7 \( 1 + (0.623 - 0.781i)T \)
11 \( 1 + (-0.433 - 0.900i)T \)
13 \( 1 + (-0.781 - 0.623i)T \)
17 \( 1 + (-0.846 + 0.532i)T \)
19 \( 1 + (-0.111 - 0.993i)T \)
23 \( 1 + (0.111 - 0.993i)T \)
29 \( 1 + (0.846 - 0.532i)T \)
31 \( 1 + (0.781 + 0.623i)T \)
37 \( 1 + (-0.330 + 0.943i)T \)
41 \( 1 + (-0.433 + 0.900i)T \)
43 \( 1 + (-0.532 - 0.846i)T \)
47 \( 1 + (0.943 - 0.330i)T \)
53 \( 1 + (0.900 + 0.433i)T \)
59 \( 1 + (0.111 + 0.993i)T \)
61 \( 1 + (0.433 + 0.900i)T \)
67 \( 1 + (-0.943 - 0.330i)T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 + (0.330 + 0.943i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.532 - 0.846i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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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

−29.580459469395202319507782370938, −28.61361083331639526641229219635, −27.69206805213095055923563874825, −26.62369274287604516162127323859, −25.27557847680823716121778571563, −24.66338506255520560753655744488, −23.714692863898099516524572613189, −22.598993790885468723973375580826, −21.76924034861131201335137794514, −21.000332020629509614633418286346, −18.76742016628884251283387286138, −17.781915827407028859523794003560, −17.39560746853647709746954398326, −16.17031383911686613581075407175, −15.10995764136001331702474648957, −13.944548989631672408383731781670, −12.74724641570723138718076747780, −11.86697761682211331856810482852, −10.13385119603816014960240269896, −9.113787224436347935043007992684, −7.51964143237746380023610118864, −6.408704297237204032016585408550, −5.32659177074586520220609972026, −4.64753711159235700294071193913, −2.01539412280955938837233190779, 0.949567376598787227790140589049, 2.59298235917792855028580907021, 4.470236842629748317390496858423, 5.3364761790214392511375538095, 6.63210118099401132937254945588, 8.59312540959771895278868234435, 10.261233291374356108483547842196, 10.5561622355538510063569362744, 11.73312544492821552999013262943, 13.06206927752930216095767551956, 13.76260716564158515640269740240, 15.16245833192390134533320653749, 16.98182672729563402601825401785, 17.58273067200796675217158622213, 18.473816403432706157098865790850, 19.87221337853523518945429890104, 21.07904329608572730084881706093, 21.74935394925690547002014635311, 22.56667942223332108667996897226, 23.76482876070593720452355996582, 24.505002672467616714631279452609, 26.5519679532657583476459379495, 27.04736526814587317470035887681, 28.43010611152880094275485869320, 28.99732712550161271103255093445

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