Properties

Label 2-10e2-100.91-c0-0-0
Degree $2$
Conductor $100$
Sign $0.728 + 0.684i$
Analytic cond. $0.0499065$
Root an. cond. $0.223397$
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  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)5-s + (0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (−0.809 + 0.587i)10-s + (−0.5 + 0.363i)13-s + (−0.809 + 0.587i)16-s + (−0.5 + 1.53i)17-s + 18-s + 20-s + (−0.809 − 0.587i)25-s + 0.618·26-s + (−0.5 − 1.53i)29-s + 32-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)5-s + (0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (−0.809 + 0.587i)10-s + (−0.5 + 0.363i)13-s + (−0.809 + 0.587i)16-s + (−0.5 + 1.53i)17-s + 18-s + 20-s + (−0.809 − 0.587i)25-s + 0.618·26-s + (−0.5 − 1.53i)29-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.728 + 0.684i$
Analytic conductor: \(0.0499065\)
Root analytic conductor: \(0.223397\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :0),\ 0.728 + 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4075006026\)
\(L(\frac12)\) \(\approx\) \(0.4075006026\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (0.809 - 0.587i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.809 + 0.587i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.66748595185211050043441405446, −12.80964274905523418157293362557, −11.80942225746427584641422958691, −10.78338008992297043504455948009, −9.621676144300964856013990338315, −8.655325615439447738633677550011, −7.79299994935581846419386074728, −5.99243829011274036691812126531, −4.26746565859084420235790164286, −2.16566075758153450204676947752, 2.77583471385176013367479650464, 5.32983035558480357806833535857, 6.55853129117874277206299100115, 7.46697379647670625499317743355, 8.900279530223792754793762110576, 9.806097530092695307679460379295, 10.91836552352720941025338915605, 11.79533847394713679437616359153, 13.61194365105627545176539506433, 14.54597469732440986161986399890

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