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Properties

Label 1-107-107.37-r0-0-0
Degree $1$
Conductor $107$
Sign $0.495 + 0.868i$
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  + (−0.630 + 0.776i)2-s + (0.829 + 0.558i)3-s + (−0.205 − 0.978i)4-s + (0.582 − 0.812i)5-s + (−0.956 + 0.292i)6-s + (−0.861 + 0.508i)7-s + (0.889 + 0.456i)8-s + (0.375 + 0.926i)9-s + (0.263 + 0.964i)10-s + (0.757 − 0.652i)11-s + (0.375 − 0.926i)12-s + (0.992 + 0.118i)13-s + (0.147 − 0.989i)14-s + (0.937 − 0.348i)15-s + (−0.915 + 0.403i)16-s + (−0.794 + 0.606i)17-s + ⋯
L(s)  = 1  + (−0.630 + 0.776i)2-s + (0.829 + 0.558i)3-s + (−0.205 − 0.978i)4-s + (0.582 − 0.812i)5-s + (−0.956 + 0.292i)6-s + (−0.861 + 0.508i)7-s + (0.889 + 0.456i)8-s + (0.375 + 0.926i)9-s + (0.263 + 0.964i)10-s + (0.757 − 0.652i)11-s + (0.375 − 0.926i)12-s + (0.992 + 0.118i)13-s + (0.147 − 0.989i)14-s + (0.937 − 0.348i)15-s + (−0.915 + 0.403i)16-s + (−0.794 + 0.606i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $0.495 + 0.868i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ 0.495 + 0.868i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8856792124 + 0.5142003524i\)
\(L(\frac12)\) \(\approx\) \(0.8856792124 + 0.5142003524i\)
\(L(1)\) \(\approx\) \(0.9466628701 + 0.3974074506i\)
\(L(1)\) \(\approx\) \(0.9466628701 + 0.3974074506i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (-0.630 + 0.776i)T \)
3 \( 1 + (0.829 + 0.558i)T \)
5 \( 1 + (0.582 - 0.812i)T \)
7 \( 1 + (-0.861 + 0.508i)T \)
11 \( 1 + (0.757 - 0.652i)T \)
13 \( 1 + (0.992 + 0.118i)T \)
17 \( 1 + (-0.794 + 0.606i)T \)
19 \( 1 + (0.972 - 0.234i)T \)
23 \( 1 + (0.147 + 0.989i)T \)
29 \( 1 + (-0.984 + 0.176i)T \)
31 \( 1 + (-0.430 - 0.902i)T \)
37 \( 1 + (-0.717 - 0.696i)T \)
41 \( 1 + (-0.533 + 0.845i)T \)
43 \( 1 + (0.582 + 0.812i)T \)
47 \( 1 + (-0.533 - 0.845i)T \)
53 \( 1 + (-0.630 - 0.776i)T \)
59 \( 1 + (-0.984 - 0.176i)T \)
61 \( 1 + (-0.861 - 0.508i)T \)
67 \( 1 + (0.889 - 0.456i)T \)
71 \( 1 + (0.829 - 0.558i)T \)
73 \( 1 + (0.0296 - 0.999i)T \)
79 \( 1 + (0.674 + 0.737i)T \)
83 \( 1 + (0.482 + 0.875i)T \)
89 \( 1 + (-0.956 - 0.292i)T \)
97 \( 1 + (0.263 + 0.964i)T \)
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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

−29.4155671852617096621778621672, −28.81794102573329586765334031532, −27.225374853770909810180255098818, −26.24374411843649362867908246906, −25.71128822194728221486127373278, −24.77991412281453736500172471594, −22.88761533105686435003873982195, −22.21128578454649075839126000288, −20.65758346922955743295325674805, −20.112768418159382988556304482288, −18.93070853657592960487447596011, −18.267319231945138344705073423889, −17.237916352441721693854707949401, −15.732928626589893733142143181697, −14.11383575135327622065384562551, −13.400730496149111619800493368298, −12.293475164764189599775101814263, −10.8425295845357384950447196246, −9.70704721036777506538962871338, −8.9345156856271297922463657160, −7.32573386995443113861133832098, −6.59174588878967518837892429440, −3.822288380356785581841859935606, −2.86900421045087739907954761433, −1.51875495977907129181984076651, 1.69967061647271132891922246309, 3.697768324470577715677779414309, 5.33477175366355345388524030530, 6.40961445236561531218994493320, 8.122626099218977583741467467586, 9.21741282945142220949937556318, 9.44653300380601351099790470003, 11.0665827290697221556980360013, 13.16988786285016895908674992805, 13.8638378035757291912986855006, 15.25010380719439111354111107078, 16.09664162493989307468450102375, 16.8436257285460411119132152212, 18.26142421776779038888690820237, 19.417627898075782597314246862084, 20.15181322741681316757194851817, 21.49027934231058143005601983451, 22.49117609450681855823380504633, 24.166615595395954725492674229782, 24.88436071988635152090418553750, 25.756250571620730635996312522476, 26.42283248119536550810570508939, 27.75733285086179465706741482006, 28.36089910908143600322831447971, 29.522045907878397855281320798196

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