Properties

Label 1-107-107.31-r1-0-0
Degree $1$
Conductor $107$
Sign $0.764 - 0.644i$
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  + (−0.0296 + 0.999i)2-s + (0.482 − 0.875i)3-s + (−0.998 − 0.0592i)4-s + (0.984 − 0.176i)5-s + (0.861 + 0.508i)6-s + (0.956 − 0.292i)7-s + (0.0887 − 0.996i)8-s + (−0.533 − 0.845i)9-s + (0.147 + 0.989i)10-s + (−0.794 − 0.606i)11-s + (−0.533 + 0.845i)12-s + (−0.915 − 0.403i)13-s + (0.263 + 0.964i)14-s + (0.320 − 0.947i)15-s + (0.992 + 0.118i)16-s + (−0.757 + 0.652i)17-s + ⋯
L(s)  = 1  + (−0.0296 + 0.999i)2-s + (0.482 − 0.875i)3-s + (−0.998 − 0.0592i)4-s + (0.984 − 0.176i)5-s + (0.861 + 0.508i)6-s + (0.956 − 0.292i)7-s + (0.0887 − 0.996i)8-s + (−0.533 − 0.845i)9-s + (0.147 + 0.989i)10-s + (−0.794 − 0.606i)11-s + (−0.533 + 0.845i)12-s + (−0.915 − 0.403i)13-s + (0.263 + 0.964i)14-s + (0.320 − 0.947i)15-s + (0.992 + 0.118i)16-s + (−0.757 + 0.652i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $0.764 - 0.644i$
Analytic conductor: \(11.4987\)
Root analytic conductor: \(11.4987\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (1:\ ),\ 0.764 - 0.644i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.872219821 - 0.6843999676i\)
\(L(\frac12)\) \(\approx\) \(1.872219821 - 0.6843999676i\)
\(L(1)\) \(\approx\) \(1.323250485 - 0.05813009702i\)
\(L(1)\) \(\approx\) \(1.323250485 - 0.05813009702i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (-0.0296 + 0.999i)T \)
3 \( 1 + (0.482 - 0.875i)T \)
5 \( 1 + (0.984 - 0.176i)T \)
7 \( 1 + (0.956 - 0.292i)T \)
11 \( 1 + (-0.794 - 0.606i)T \)
13 \( 1 + (-0.915 - 0.403i)T \)
17 \( 1 + (-0.757 + 0.652i)T \)
19 \( 1 + (0.674 - 0.737i)T \)
23 \( 1 + (0.263 - 0.964i)T \)
29 \( 1 + (0.582 + 0.812i)T \)
31 \( 1 + (0.717 - 0.696i)T \)
37 \( 1 + (-0.430 - 0.902i)T \)
41 \( 1 + (0.375 + 0.926i)T \)
43 \( 1 + (0.984 + 0.176i)T \)
47 \( 1 + (0.375 - 0.926i)T \)
53 \( 1 + (0.0296 + 0.999i)T \)
59 \( 1 + (-0.582 + 0.812i)T \)
61 \( 1 + (-0.956 - 0.292i)T \)
67 \( 1 + (0.0887 + 0.996i)T \)
71 \( 1 + (-0.482 - 0.875i)T \)
73 \( 1 + (0.630 + 0.776i)T \)
79 \( 1 + (0.972 - 0.234i)T \)
83 \( 1 + (0.829 + 0.558i)T \)
89 \( 1 + (-0.861 + 0.508i)T \)
97 \( 1 + (-0.147 - 0.989i)T \)
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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

−29.24900065125275200056073386420, −28.616400701051303005373046055474, −27.390659321201507757022461788500, −26.71608273763688556781568976524, −25.66194159624800203219897427534, −24.460261244858755979059326628671, −22.77915399061318860086204869233, −21.88150941827683182842017706086, −21.026041909437924685539142228933, −20.55634845061459050651937090734, −19.20420617317755710141176200332, −17.93880274247863251154020446634, −17.2329090210440645409795413215, −15.47157607482337804239203334534, −14.26991655279550574413670511189, −13.656380552330874777619436345590, −12.08253588976084375086939877883, −10.85449053157077714870376442382, −9.903387167369193663240631149116, −9.14169757169607278147085473377, −7.7907291699697907854507214160, −5.31338935860172499687999314502, −4.61854532560516428269393412358, −2.81460794546568831296271178704, −1.916923408505728202687418870720, 0.84786238408228139026437824106, 2.56501814321065607904903602381, 4.767178680744083399512469745885, 5.902000852239760487089380759588, 7.14397992206040902859388887549, 8.17799538893764946810304459029, 9.10378310749554499270722243020, 10.599355140988639035805673584889, 12.55739970605704456887637751992, 13.506683838723554648892300124540, 14.22205906469130535458047836809, 15.23142610479651950736736679041, 16.86336673904005668938049989227, 17.74994943233699303407894271227, 18.294914762424344456804175266806, 19.7149319906587897906316298911, 21.01933236654953647647413042875, 22.119420261712801529554654206569, 23.55673223214180792235820462544, 24.5344709536987748974063290133, 24.70822054814948010828796681187, 26.21001880892251522218892488277, 26.62904528375657248345565357313, 28.24717160526628244007098423719, 29.331919833698356109539297266037

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