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Properties

Label 2-20e2-16.5-c1-0-14
Degree $2$
Conductor $400$
Sign $0.917 - 0.397i$
Analytic cond. $3.19401$
Root an. cond. $1.78718$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.17 − 0.790i)2-s + (−1.37 + 1.37i)3-s + (0.750 − 1.85i)4-s + (−0.523 + 2.68i)6-s + 2.73i·7-s + (−0.584 − 2.76i)8-s − 0.755i·9-s + (4.12 + 4.12i)11-s + (1.51 + 3.56i)12-s + (1.37 − 1.37i)13-s + (2.16 + 3.20i)14-s + (−2.87 − 2.78i)16-s + 4.94·17-s + (−0.596 − 0.885i)18-s + (−0.292 + 0.292i)19-s + ⋯
L(s)  = 1  + (0.829 − 0.558i)2-s + (−0.791 + 0.791i)3-s + (0.375 − 0.926i)4-s + (−0.213 + 1.09i)6-s + 1.03i·7-s + (−0.206 − 0.978i)8-s − 0.251i·9-s + (1.24 + 1.24i)11-s + (0.436 + 1.03i)12-s + (0.382 − 0.382i)13-s + (0.577 + 0.857i)14-s + (−0.718 − 0.695i)16-s + 1.20·17-s + (−0.140 − 0.208i)18-s + (−0.0671 + 0.0671i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.917 - 0.397i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.917 - 0.397i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73375 + 0.359175i\)
\(L(\frac12)\) \(\approx\) \(1.73375 + 0.359175i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.17 + 0.790i)T \)
5 \( 1 \)
good3 \( 1 + (1.37 - 1.37i)T - 3iT^{2} \)
7 \( 1 - 2.73iT - 7T^{2} \)
11 \( 1 + (-4.12 - 4.12i)T + 11iT^{2} \)
13 \( 1 + (-1.37 + 1.37i)T - 13iT^{2} \)
17 \( 1 - 4.94T + 17T^{2} \)
19 \( 1 + (0.292 - 0.292i)T - 19iT^{2} \)
23 \( 1 - 1.64iT - 23T^{2} \)
29 \( 1 + (5.67 - 5.67i)T - 29iT^{2} \)
31 \( 1 - 3.95T + 31T^{2} \)
37 \( 1 + (2.48 + 2.48i)T + 37iT^{2} \)
41 \( 1 + 8.40iT - 41T^{2} \)
43 \( 1 + (-3.22 - 3.22i)T + 43iT^{2} \)
47 \( 1 - 5.19T + 47T^{2} \)
53 \( 1 + (7.20 + 7.20i)T + 53iT^{2} \)
59 \( 1 + (6.41 + 6.41i)T + 59iT^{2} \)
61 \( 1 + (3.82 - 3.82i)T - 61iT^{2} \)
67 \( 1 + (5.76 - 5.76i)T - 67iT^{2} \)
71 \( 1 + 7.92iT - 71T^{2} \)
73 \( 1 + 4.36iT - 73T^{2} \)
79 \( 1 + 5.56T + 79T^{2} \)
83 \( 1 + (-0.516 + 0.516i)T - 83iT^{2} \)
89 \( 1 + 6.42iT - 89T^{2} \)
97 \( 1 - 9.44T + 97T^{2} \)
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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

−11.46645014695427634561878319976, −10.58110366859138515759281089067, −9.772552383987584912460324088850, −9.050722280982362884756699818542, −7.33572398990398855661036054725, −6.06010504437044713013251193780, −5.42255526539953658731268283369, −4.49523271955328721118960093853, −3.42988896995041762446495027126, −1.79054081679361484003474689907, 1.14631418595028586028319466635, 3.38598971274382289952914261307, 4.29646001442211906513470597219, 5.82483041594957775840196575138, 6.29813956924835425344290786038, 7.18364648633092645208257666542, 8.034756142924629898488379022312, 9.235489074350300973771223941688, 10.74266599701970910816718696075, 11.56261496524716646388383463263

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