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Properties

Label 1-107-107.32-r1-0-0
Degree $1$
Conductor $107$
Sign $-0.552 + 0.833i$
Analytic cond. $11.4987$
Root an. cond. $11.4987$
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  + (0.956 + 0.292i)2-s + (−0.320 + 0.947i)3-s + (0.829 + 0.558i)4-s + (0.205 + 0.978i)5-s + (−0.582 + 0.812i)6-s + (0.984 + 0.176i)7-s + (0.630 + 0.776i)8-s + (−0.794 − 0.606i)9-s + (−0.0887 + 0.996i)10-s + (0.972 + 0.234i)11-s + (−0.794 + 0.606i)12-s + (−0.533 − 0.845i)13-s + (0.889 + 0.456i)14-s + (−0.992 − 0.118i)15-s + (0.375 + 0.926i)16-s + (−0.674 + 0.737i)17-s + ⋯
L(s)  = 1  + (0.956 + 0.292i)2-s + (−0.320 + 0.947i)3-s + (0.829 + 0.558i)4-s + (0.205 + 0.978i)5-s + (−0.582 + 0.812i)6-s + (0.984 + 0.176i)7-s + (0.630 + 0.776i)8-s + (−0.794 − 0.606i)9-s + (−0.0887 + 0.996i)10-s + (0.972 + 0.234i)11-s + (−0.794 + 0.606i)12-s + (−0.533 − 0.845i)13-s + (0.889 + 0.456i)14-s + (−0.992 − 0.118i)15-s + (0.375 + 0.926i)16-s + (−0.674 + 0.737i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $-0.552 + 0.833i$
Analytic conductor: \(11.4987\)
Root analytic conductor: \(11.4987\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (1:\ ),\ -0.552 + 0.833i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.480367990 + 2.757853056i\)
\(L(\frac12)\) \(\approx\) \(1.480367990 + 2.757853056i\)
\(L(1)\) \(\approx\) \(1.505819539 + 1.228666298i\)
\(L(1)\) \(\approx\) \(1.505819539 + 1.228666298i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (0.956 + 0.292i)T \)
3 \( 1 + (-0.320 + 0.947i)T \)
5 \( 1 + (0.205 + 0.978i)T \)
7 \( 1 + (0.984 + 0.176i)T \)
11 \( 1 + (0.972 + 0.234i)T \)
13 \( 1 + (-0.533 - 0.845i)T \)
17 \( 1 + (-0.674 + 0.737i)T \)
19 \( 1 + (-0.430 - 0.902i)T \)
23 \( 1 + (0.889 - 0.456i)T \)
29 \( 1 + (-0.998 - 0.0592i)T \)
31 \( 1 + (-0.147 + 0.989i)T \)
37 \( 1 + (0.263 - 0.964i)T \)
41 \( 1 + (0.757 - 0.652i)T \)
43 \( 1 + (0.205 - 0.978i)T \)
47 \( 1 + (0.757 + 0.652i)T \)
53 \( 1 + (-0.956 + 0.292i)T \)
59 \( 1 + (0.998 - 0.0592i)T \)
61 \( 1 + (-0.984 + 0.176i)T \)
67 \( 1 + (0.630 - 0.776i)T \)
71 \( 1 + (0.320 + 0.947i)T \)
73 \( 1 + (0.861 - 0.508i)T \)
79 \( 1 + (-0.717 - 0.696i)T \)
83 \( 1 + (0.937 - 0.348i)T \)
89 \( 1 + (0.582 + 0.812i)T \)
97 \( 1 + (0.0887 - 0.996i)T \)
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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

−29.35623649258198597165388718603, −28.32828185895530018857156516556, −27.33750400968899724952130085928, −25.286628013285525844183867730910, −24.47529452948422568208356539198, −24.075718850219803623915651956263, −22.95636847502647268022806202800, −21.796783804963287479438208154715, −20.7232935302178944352152854348, −19.83155488463010333759401827997, −18.74421920980687463596760867635, −17.20406556121470923748747184395, −16.555056590149260539156145154512, −14.74539032665747590573245753359, −13.85238434623748434472505462233, −12.92661462406550223389219933846, −11.80247544647061540775621143403, −11.26195984690688529279727816708, −9.29227648710579919755581367990, −7.76494845605190402867891726196, −6.506711802778843341788490719688, −5.28618112188823321968551363827, −4.26064712808620321717148502449, −2.08947273372657437217601358083, −1.13162178625413018491282992899, 2.38165670756132080022260262044, 3.78693937232300005757011735561, 4.90897202244481755249386802130, 6.06284574669029747027885742648, 7.27417775275322095573774537658, 8.9206253796616530242658316014, 10.70535485393624916068298160889, 11.174460364743171492092357295021, 12.49972151006284267050262890775, 14.176969343643987830547967115, 14.8893024490811483325837825534, 15.46524570481669521766771548313, 17.15236529244726171650652297215, 17.62594634837129611601946496180, 19.63763216168127782767474048213, 20.762848651379203418683958317753, 21.83112048341331259220676704300, 22.25515835897281633315383805515, 23.271074808534613457509631299230, 24.50770974245559300057309414855, 25.56913023747238124299349767048, 26.62892961615537066523047368808, 27.5090497720223253684214562978, 28.79705267015565120192001350451, 30.122290276164034770284642959149

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