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Properties

Label 1-113-113.106-r0-0-0
Degree $1$
Conductor $113$
Sign $-0.781 + 0.624i$
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 + (−0.900 − 0.433i)3-s + (−0.222 − 0.974i)4-s + (−0.900 + 0.433i)5-s + (−0.900 + 0.433i)6-s + (−0.900 − 0.433i)7-s + (−0.900 − 0.433i)8-s + (0.623 + 0.781i)9-s + (−0.222 + 0.974i)10-s + (−0.222 + 0.974i)11-s + (−0.222 + 0.974i)12-s + (−0.900 − 0.433i)13-s + (−0.900 + 0.433i)14-s + 15-s + (−0.900 + 0.433i)16-s + (0.623 − 0.781i)17-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.900 − 0.433i)3-s + (−0.222 − 0.974i)4-s + (−0.900 + 0.433i)5-s + (−0.900 + 0.433i)6-s + (−0.900 − 0.433i)7-s + (−0.900 − 0.433i)8-s + (0.623 + 0.781i)9-s + (−0.222 + 0.974i)10-s + (−0.222 + 0.974i)11-s + (−0.222 + 0.974i)12-s + (−0.900 − 0.433i)13-s + (−0.900 + 0.433i)14-s + 15-s + (−0.900 + 0.433i)16-s + (0.623 − 0.781i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $-0.781 + 0.624i$
Analytic conductor: \(0.524769\)
Root analytic conductor: \(0.524769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (0:\ ),\ -0.781 + 0.624i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09885924082 - 0.2821989600i\)
\(L(\frac12)\) \(\approx\) \(-0.09885924082 - 0.2821989600i\)
\(L(1)\) \(\approx\) \(0.4645499803 - 0.3978726509i\)
\(L(1)\) \(\approx\) \(0.4645499803 - 0.3978726509i\)

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 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 + (-0.900 - 0.433i)T \)
11 \( 1 + (-0.222 + 0.974i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
17 \( 1 + (0.623 - 0.781i)T \)
19 \( 1 + (-0.900 - 0.433i)T \)
23 \( 1 + (-0.900 + 0.433i)T \)
29 \( 1 + (0.623 - 0.781i)T \)
31 \( 1 + (-0.900 - 0.433i)T \)
37 \( 1 + (-0.222 + 0.974i)T \)
41 \( 1 + (-0.222 - 0.974i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (-0.222 - 0.974i)T \)
59 \( 1 + (-0.900 - 0.433i)T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 + (-0.222 + 0.974i)T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 + (-0.222 - 0.974i)T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
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

−29.94368042182386483418210506956, −28.98638317697051271469900375985, −27.796748157790726113740268433326, −26.90555363503634924981424842782, −25.97165783310573566818192405055, −24.536790308283863826733518110574, −23.72725894445696080694340290078, −23.00425611987626052395083247102, −21.90975525030028542851548654780, −21.30635131881665565738410453557, −19.61146941842987946682431534115, −18.41330032394946536356654722240, −16.83031645317258728421050844361, −16.38963132739276719414163287583, −15.548604024714972848029895958009, −14.43665599230690725842828934413, −12.52901353967613337567401750876, −12.38642367713846418212433042506, −10.90941153788221285406579039152, −9.27303400046656981012243083220, −8.02123443169480325997420682821, −6.58352680360219945318476571921, −5.64678357948304914645530759239, −4.41192573556716071554208704480, −3.35230045186755106256026860422, 0.25047063212741963615366222664, 2.40883381593720241025498784143, 3.93430207967457722443101290519, 5.10318054643812079210681896833, 6.576474958346517254315345401518, 7.51212082083705011183021181716, 9.84776025828510889857505851126, 10.582413496316692211765090852883, 11.89495406677145348632817733370, 12.43629759157566605996373779614, 13.549033679623863978677127790156, 15.01618424143933644704098656578, 15.987298411637853233942624179501, 17.43232586604424995487402232845, 18.65518877106716699478543558531, 19.4502311317134358239507377828, 20.32520634909348413086049208563, 21.9575262636354295581364102055, 22.69852724198076977230913928382, 23.28586886826267690197252958058, 24.130229976292003155761137985127, 25.6306914201383265226058084230, 27.22772601548752396448172590970, 27.881472613718377035957250240786, 29.00078346035219613469296432627

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