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Properties

Label 1-107-107.105-r0-0-0
Degree $1$
Conductor $107$
Sign $0.457 - 0.888i$
Analytic cond. $0.496905$
Root an. cond. $0.496905$
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  + (−0.998 − 0.0592i)2-s + (−0.533 − 0.845i)3-s + (0.992 + 0.118i)4-s + (0.937 − 0.348i)5-s + (0.482 + 0.875i)6-s + (0.829 − 0.558i)7-s + (−0.984 − 0.176i)8-s + (−0.430 + 0.902i)9-s + (−0.956 + 0.292i)10-s + (0.263 + 0.964i)11-s + (−0.430 − 0.902i)12-s + (0.674 + 0.737i)13-s + (−0.861 + 0.508i)14-s + (−0.794 − 0.606i)15-s + (0.972 + 0.234i)16-s + (0.147 − 0.989i)17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0592i)2-s + (−0.533 − 0.845i)3-s + (0.992 + 0.118i)4-s + (0.937 − 0.348i)5-s + (0.482 + 0.875i)6-s + (0.829 − 0.558i)7-s + (−0.984 − 0.176i)8-s + (−0.430 + 0.902i)9-s + (−0.956 + 0.292i)10-s + (0.263 + 0.964i)11-s + (−0.430 − 0.902i)12-s + (0.674 + 0.737i)13-s + (−0.861 + 0.508i)14-s + (−0.794 − 0.606i)15-s + (0.972 + 0.234i)16-s + (0.147 − 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.457 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.457 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $0.457 - 0.888i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ 0.457 - 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6164257649 - 0.3758581321i\)
\(L(\frac12)\) \(\approx\) \(0.6164257649 - 0.3758581321i\)
\(L(1)\) \(\approx\) \(0.7033422302 - 0.2557641638i\)
\(L(1)\) \(\approx\) \(0.7033422302 - 0.2557641638i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (-0.998 - 0.0592i)T \)
3 \( 1 + (-0.533 - 0.845i)T \)
5 \( 1 + (0.937 - 0.348i)T \)
7 \( 1 + (0.829 - 0.558i)T \)
11 \( 1 + (0.263 + 0.964i)T \)
13 \( 1 + (0.674 + 0.737i)T \)
17 \( 1 + (0.147 - 0.989i)T \)
19 \( 1 + (-0.0887 - 0.996i)T \)
23 \( 1 + (-0.861 - 0.508i)T \)
29 \( 1 + (-0.320 + 0.947i)T \)
31 \( 1 + (0.0296 - 0.999i)T \)
37 \( 1 + (-0.630 + 0.776i)T \)
41 \( 1 + (-0.717 + 0.696i)T \)
43 \( 1 + (0.937 + 0.348i)T \)
47 \( 1 + (-0.717 - 0.696i)T \)
53 \( 1 + (-0.998 + 0.0592i)T \)
59 \( 1 + (-0.320 - 0.947i)T \)
61 \( 1 + (0.829 + 0.558i)T \)
67 \( 1 + (-0.984 + 0.176i)T \)
71 \( 1 + (-0.533 + 0.845i)T \)
73 \( 1 + (-0.205 + 0.978i)T \)
79 \( 1 + (0.889 - 0.456i)T \)
83 \( 1 + (0.375 + 0.926i)T \)
89 \( 1 + (0.482 - 0.875i)T \)
97 \( 1 + (-0.956 + 0.292i)T \)
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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

−29.526376413096748654615310502156, −28.51483857640230811980182007293, −27.73004109453442407904122097647, −26.90317072372216947243686721816, −25.88318665732653223612145378385, −24.965218121711517004940529229581, −23.83006282921252889199316030328, −22.2635712889275610501464554324, −21.25839650668272547139037203312, −20.77360220046296434335217862984, −19.10211406597098908650367685804, −17.945999911749067688048411218673, −17.39193991926430377168716730015, −16.27002448890765643936811146283, −15.204692943108227721507906083242, −14.185773906420664345994056610289, −12.16574367209966244225955946066, −10.954249623964858991444318120, −10.355707911488651584354567571145, −9.111030046383538797088934948961, −8.135710566746853713558317504077, −6.0917430276722056591925858485, −5.66222285099428261214275392422, −3.40829058900764277136358212072, −1.6401249323484179750186185279, 1.258576809761804593398115503558, 2.1938988358830094445732178221, 4.89300191582775697413834338101, 6.406039722712832398389122960900, 7.27924198649171774896690267286, 8.56571090946017593110776430032, 9.79984243832565248212771684198, 11.04015692428073630702969270075, 11.953325739561595451224911658391, 13.28093084382415639684592154910, 14.42538771346298667389214917116, 16.2441879527062908942944284517, 17.17201294806989353451691333934, 17.86290037034408329155605272558, 18.576073759946348482055741780, 20.08157676339654060877333467533, 20.768369811911682116262172673613, 22.12655613377944987740729257173, 23.69048192862706363749151827697, 24.41647090618174304022548993917, 25.39272209557716054453139708974, 26.22542561112237605042827768931, 27.820737889645813668707105125916, 28.28705849423604311535961139490, 29.39597250315226240727566599515

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