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Properties

Label 1-107-107.52-r0-0-0
Degree $1$
Conductor $107$
Sign $-0.566 + 0.823i$
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  + (0.582 + 0.812i)2-s + (−0.915 − 0.403i)3-s + (−0.320 + 0.947i)4-s + (0.829 + 0.558i)5-s + (−0.205 − 0.978i)6-s + (−0.998 + 0.0592i)7-s + (−0.956 + 0.292i)8-s + (0.674 + 0.737i)9-s + (0.0296 + 0.999i)10-s + (−0.430 + 0.902i)11-s + (0.674 − 0.737i)12-s + (0.757 + 0.652i)13-s + (−0.630 − 0.776i)14-s + (−0.533 − 0.845i)15-s + (−0.794 − 0.606i)16-s + (−0.717 + 0.696i)17-s + ⋯
L(s)  = 1  + (0.582 + 0.812i)2-s + (−0.915 − 0.403i)3-s + (−0.320 + 0.947i)4-s + (0.829 + 0.558i)5-s + (−0.205 − 0.978i)6-s + (−0.998 + 0.0592i)7-s + (−0.956 + 0.292i)8-s + (0.674 + 0.737i)9-s + (0.0296 + 0.999i)10-s + (−0.430 + 0.902i)11-s + (0.674 − 0.737i)12-s + (0.757 + 0.652i)13-s + (−0.630 − 0.776i)14-s + (−0.533 − 0.845i)15-s + (−0.794 − 0.606i)16-s + (−0.717 + 0.696i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $-0.566 + 0.823i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ -0.566 + 0.823i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4381609667 + 0.8331934601i\)
\(L(\frac12)\) \(\approx\) \(0.4381609667 + 0.8331934601i\)
\(L(1)\) \(\approx\) \(0.8027133218 + 0.6020108076i\)
\(L(1)\) \(\approx\) \(0.8027133218 + 0.6020108076i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (0.582 + 0.812i)T \)
3 \( 1 + (-0.915 - 0.403i)T \)
5 \( 1 + (0.829 + 0.558i)T \)
7 \( 1 + (-0.998 + 0.0592i)T \)
11 \( 1 + (-0.430 + 0.902i)T \)
13 \( 1 + (0.757 + 0.652i)T \)
17 \( 1 + (-0.717 + 0.696i)T \)
19 \( 1 + (0.147 - 0.989i)T \)
23 \( 1 + (-0.630 + 0.776i)T \)
29 \( 1 + (0.482 - 0.875i)T \)
31 \( 1 + (0.889 + 0.456i)T \)
37 \( 1 + (-0.0887 - 0.996i)T \)
41 \( 1 + (0.972 + 0.234i)T \)
43 \( 1 + (0.829 - 0.558i)T \)
47 \( 1 + (0.972 - 0.234i)T \)
53 \( 1 + (0.582 - 0.812i)T \)
59 \( 1 + (0.482 + 0.875i)T \)
61 \( 1 + (-0.998 - 0.0592i)T \)
67 \( 1 + (-0.956 - 0.292i)T \)
71 \( 1 + (-0.915 + 0.403i)T \)
73 \( 1 + (-0.984 - 0.176i)T \)
79 \( 1 + (0.263 - 0.964i)T \)
83 \( 1 + (0.992 + 0.118i)T \)
89 \( 1 + (-0.205 + 0.978i)T \)
97 \( 1 + (0.0296 + 0.999i)T \)
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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

−29.24863646637424751631722914693, −28.5937389395149518470228553492, −27.64450722422901784738178675668, −26.42738713441565567210469865413, −24.89265977653758302544087939664, −23.84111205633355017021841665154, −22.71691864577155794454175058055, −22.11882697294190715868554197715, −21.03998159195941562950579055475, −20.313983636544700715739182715169, −18.74784689540974953179979272780, −17.88818752099837038292713062168, −16.42229547643697002164050066444, −15.70451802035703036306392342564, −13.90854249099447499494256100271, −13.026786504048016670295869862684, −12.1446948613884639482967793462, −10.74460795620968423033148603426, −10.04693178677441421400093533833, −8.9084449990501413642349815323, −6.24548233271280970062194596147, −5.72232749394372984258694924463, −4.37361089099763884243064434735, −2.930196874128619747970555245002, −0.89311274420787443014402894395, 2.3877445547969797007012814680, 4.246049251698583939376479443436, 5.735059112106448465361202494250, 6.47342492376679981235704639184, 7.323763734737222046919743209004, 9.20810922390876184299961524183, 10.53137586084057217922328848261, 11.97348523110048322954353473151, 13.142766863022472840022387439505, 13.68496292566934083315233747413, 15.35778088774875638273082883715, 16.163243405281750595120938420101, 17.49559137602452815417230060002, 17.917761090040994247395554872388, 19.270706438620378703407390965884, 21.208939017311604532684379507269, 22.0433129081749399265814923673, 22.851948542962200436425238150118, 23.62001236883834246158465621249, 24.80334404189957553311470315676, 25.82698184225924627177567905916, 26.397427432610179121196954313738, 28.23489913475926005037784212620, 28.959643186040101925876481099363, 30.16321221884889209440403123415

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