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Properties

Label 1-107-107.19-r0-0-0
Degree $1$
Conductor $107$
Sign $0.309 + 0.950i$
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  + (−0.0887 − 0.996i)2-s + (−0.998 − 0.0592i)3-s + (−0.984 + 0.176i)4-s + (−0.861 − 0.508i)5-s + (0.0296 + 0.999i)6-s + (−0.630 − 0.776i)7-s + (0.263 + 0.964i)8-s + (0.992 + 0.118i)9-s + (−0.430 + 0.902i)10-s + (0.375 + 0.926i)11-s + (0.992 − 0.118i)12-s + (−0.320 + 0.947i)13-s + (−0.717 + 0.696i)14-s + (0.829 + 0.558i)15-s + (0.937 − 0.348i)16-s + (−0.533 + 0.845i)17-s + ⋯
L(s)  = 1  + (−0.0887 − 0.996i)2-s + (−0.998 − 0.0592i)3-s + (−0.984 + 0.176i)4-s + (−0.861 − 0.508i)5-s + (0.0296 + 0.999i)6-s + (−0.630 − 0.776i)7-s + (0.263 + 0.964i)8-s + (0.992 + 0.118i)9-s + (−0.430 + 0.902i)10-s + (0.375 + 0.926i)11-s + (0.992 − 0.118i)12-s + (−0.320 + 0.947i)13-s + (−0.717 + 0.696i)14-s + (0.829 + 0.558i)15-s + (0.937 − 0.348i)16-s + (−0.533 + 0.845i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $0.309 + 0.950i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ 0.309 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09740126861 + 0.07072154814i\)
\(L(\frac12)\) \(\approx\) \(0.09740126861 + 0.07072154814i\)
\(L(1)\) \(\approx\) \(0.3814998063 - 0.1741573782i\)
\(L(1)\) \(\approx\) \(0.3814998063 - 0.1741573782i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (-0.0887 - 0.996i)T \)
3 \( 1 + (-0.998 - 0.0592i)T \)
5 \( 1 + (-0.861 - 0.508i)T \)
7 \( 1 + (-0.630 - 0.776i)T \)
11 \( 1 + (0.375 + 0.926i)T \)
13 \( 1 + (-0.320 + 0.947i)T \)
17 \( 1 + (-0.533 + 0.845i)T \)
19 \( 1 + (-0.794 + 0.606i)T \)
23 \( 1 + (-0.717 - 0.696i)T \)
29 \( 1 + (-0.956 - 0.292i)T \)
31 \( 1 + (0.674 - 0.737i)T \)
37 \( 1 + (0.972 - 0.234i)T \)
41 \( 1 + (-0.915 + 0.403i)T \)
43 \( 1 + (-0.861 + 0.508i)T \)
47 \( 1 + (-0.915 - 0.403i)T \)
53 \( 1 + (-0.0887 + 0.996i)T \)
59 \( 1 + (-0.956 + 0.292i)T \)
61 \( 1 + (-0.630 + 0.776i)T \)
67 \( 1 + (0.263 - 0.964i)T \)
71 \( 1 + (-0.998 + 0.0592i)T \)
73 \( 1 + (0.889 + 0.456i)T \)
79 \( 1 + (0.757 + 0.652i)T \)
83 \( 1 + (-0.205 - 0.978i)T \)
89 \( 1 + (0.0296 - 0.999i)T \)
97 \( 1 + (-0.430 + 0.902i)T \)
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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

−29.32110343778069053758039708330, −27.996373530530803179992379463847, −27.38761446832006962819221839491, −26.487751638427954331311345913659, −25.180421417718482110358873451402, −24.18643719450765626229516510159, −23.28137525493996154133436112525, −22.29641449991430192940776371070, −21.899987447282967610242183187642, −19.58779700818465863684419474531, −18.656694212210152546722042733944, −17.80510626499296502712295314759, −16.548213802262907760470252408646, −15.71942284138002326964809526907, −15.044165842025506076379407638461, −13.382350515413276497762432124233, −12.20104991944525724644795714178, −11.049793236363404141670025224, −9.697271027641536611022615318138, −8.32406715514808488334802845668, −6.980451787857663203548413695, −6.11231262118170671473109012205, −4.939086631335514468181336867373, −3.42762917720143136151474310936, −0.14124447179007389408222445552, 1.66903578611051739160671093480, 4.05876050166273081519834285830, 4.47645971252059642724459955380, 6.446020041056304037360280421313, 7.8519407381550100203378842014, 9.465367286335761711490726785836, 10.47033276877518683726601008529, 11.59479053325840902677283729132, 12.41688825443972831305214777733, 13.23266103702365999731074410902, 14.971465862975933916096113581724, 16.6436129114749827649715014349, 17.04937165370677307683838408391, 18.513350890188731485880232957293, 19.498741904940835991471163029908, 20.316460364388026188246863452712, 21.58644034326382280214025653308, 22.706050021599822065602994555545, 23.29262593436013593091724448123, 24.24015621818589841526630554327, 26.19924992585404681840906915240, 27.07444375652301525034471883080, 28.18002673064789473227415002291, 28.57163175870183091477102451827, 29.77111186022001784396135994153

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