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Properties

Label 1-113-113.31-r0-0-0
Degree $1$
Conductor $113$
Sign $0.316 + 0.948i$
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  + (−0.623 + 0.781i)2-s + (−0.943 + 0.330i)3-s + (−0.222 − 0.974i)4-s + (0.943 + 0.330i)5-s + (0.330 − 0.943i)6-s + (−0.900 − 0.433i)7-s + (0.900 + 0.433i)8-s + (0.781 − 0.623i)9-s + (−0.846 + 0.532i)10-s + (0.974 + 0.222i)11-s + (0.532 + 0.846i)12-s + (0.433 − 0.900i)13-s + (0.900 − 0.433i)14-s − 15-s + (−0.900 + 0.433i)16-s + (0.111 + 0.993i)17-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)2-s + (−0.943 + 0.330i)3-s + (−0.222 − 0.974i)4-s + (0.943 + 0.330i)5-s + (0.330 − 0.943i)6-s + (−0.900 − 0.433i)7-s + (0.900 + 0.433i)8-s + (0.781 − 0.623i)9-s + (−0.846 + 0.532i)10-s + (0.974 + 0.222i)11-s + (0.532 + 0.846i)12-s + (0.433 − 0.900i)13-s + (0.900 − 0.433i)14-s − 15-s + (−0.900 + 0.433i)16-s + (0.111 + 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $0.316 + 0.948i$
Analytic conductor: \(0.524769\)
Root analytic conductor: \(0.524769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (0:\ ),\ 0.316 + 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5192423853 + 0.3740904435i\)
\(L(\frac12)\) \(\approx\) \(0.5192423853 + 0.3740904435i\)
\(L(1)\) \(\approx\) \(0.6176132562 + 0.2859453753i\)
\(L(1)\) \(\approx\) \(0.6176132562 + 0.2859453753i\)

Euler product

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

−29.06308883026266372453595168567, −28.416106944386589986782523463332, −27.54205930787210128180524823501, −26.154700277420766370814819595971, −25.228798290830586761337532562831, −24.19146462201235840866193983989, −22.442238829770848716982415966321, −22.117531794753158076102645789937, −21.00162386036198141653002802783, −19.69240626586508984021952046290, −18.602391921073115420700280157969, −17.91426455777374438774558528015, −16.664632584029867829228918846654, −16.281475552948085532314530797497, −13.861358299330996065793674681032, −12.90899498908288405502281649779, −11.970548480800442280753139305698, −10.98445426890181717619713940194, −9.610919691598342557529811962954, −9.02257610731877178615964929569, −7.03023027500375614538138467836, −6.04394635356019959946868419703, −4.43568992908880978544961940508, −2.54394391037421905481637806424, −1.0823276339661256799788350528, 1.320654331667884573034266187399, 3.899288244931394421277163604582, 5.76621488338263786120634363638, 6.15766141005829237138528821400, 7.402206537840988358404033172623, 9.27598155808911752552158762784, 10.05407830304567956879271647905, 10.848458507986696489031463858622, 12.61322301125781658898964586578, 13.88199180221637041931256403743, 15.11201004345974292341592316103, 16.21465297774387458246181357470, 17.14074195992509486638052151535, 17.721215748136766584559917542659, 18.86502437340147953342452482255, 20.108811823404304753067614047602, 21.660039032789479279506496371769, 22.69100412917356536429778911827, 23.25133531092823967868177299127, 24.71398670401671213912854085918, 25.55017976694757794494160462938, 26.49009516715032562188281372203, 27.524283302107194414465203555526, 28.426641721030429823043255194301, 29.3242069552456105315118072692

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