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Properties

Label 2-20e2-80.69-c1-0-10
Degree $2$
Conductor $400$
Sign $0.926 + 0.376i$
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  + (−0.388 − 1.35i)2-s + (1.03 + 1.03i)3-s + (−1.69 + 1.05i)4-s + (1.01 − 1.81i)6-s + 1.49·7-s + (2.09 + 1.89i)8-s − 0.836i·9-s + (0.423 + 0.423i)11-s + (−2.86 − 0.666i)12-s + (1.85 + 1.85i)13-s + (−0.581 − 2.03i)14-s + (1.76 − 3.58i)16-s + 6.50i·17-s + (−1.13 + 0.325i)18-s + (1.75 − 1.75i)19-s + ⋯
L(s)  = 1  + (−0.274 − 0.961i)2-s + (0.600 + 0.600i)3-s + (−0.849 + 0.528i)4-s + (0.412 − 0.742i)6-s + 0.565·7-s + (0.741 + 0.671i)8-s − 0.278i·9-s + (0.127 + 0.127i)11-s + (−0.827 − 0.192i)12-s + (0.515 + 0.515i)13-s + (−0.155 − 0.543i)14-s + (0.441 − 0.897i)16-s + 1.57i·17-s + (−0.268 + 0.0766i)18-s + (0.403 − 0.403i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.926 + 0.376i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.926 + 0.376i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43745 - 0.281329i\)
\(L(\frac12)\) \(\approx\) \(1.43745 - 0.281329i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.388 + 1.35i)T \)
5 \( 1 \)
good3 \( 1 + (-1.03 - 1.03i)T + 3iT^{2} \)
7 \( 1 - 1.49T + 7T^{2} \)
11 \( 1 + (-0.423 - 0.423i)T + 11iT^{2} \)
13 \( 1 + (-1.85 - 1.85i)T + 13iT^{2} \)
17 \( 1 - 6.50iT - 17T^{2} \)
19 \( 1 + (-1.75 + 1.75i)T - 19iT^{2} \)
23 \( 1 - 7.19T + 23T^{2} \)
29 \( 1 + (-6.57 + 6.57i)T - 29iT^{2} \)
31 \( 1 + 6.75T + 31T^{2} \)
37 \( 1 + (1.95 - 1.95i)T - 37iT^{2} \)
41 \( 1 - 7.70iT - 41T^{2} \)
43 \( 1 + (6.13 - 6.13i)T - 43iT^{2} \)
47 \( 1 + 6.65iT - 47T^{2} \)
53 \( 1 + (-5.29 + 5.29i)T - 53iT^{2} \)
59 \( 1 + (5.91 + 5.91i)T + 59iT^{2} \)
61 \( 1 + (1.43 - 1.43i)T - 61iT^{2} \)
67 \( 1 + (6.35 + 6.35i)T + 67iT^{2} \)
71 \( 1 + 4.08iT - 71T^{2} \)
73 \( 1 + 2.43T + 73T^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 + (2.81 + 2.81i)T + 83iT^{2} \)
89 \( 1 - 10.5iT - 89T^{2} \)
97 \( 1 - 18.1iT - 97T^{2} \)
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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

−11.15290518071664970371004759413, −10.29064026586161870066063585171, −9.440074360214828716196903755251, −8.686737491905607107395315999772, −8.021631683276992216954348987939, −6.52456462765559191601728580513, −4.90966861408610388692607626727, −3.99944393161576187928173641851, −3.04425135034524929405660092341, −1.51176093179486007440686666097, 1.29619591353907710448082811289, 3.10730947189431361743111001792, 4.80159018221084952388293610179, 5.57291173875312302066584980806, 7.10301299945005382871220399736, 7.40043197894823449261495452424, 8.587678958277478026565546335974, 8.982471113062198987590161079393, 10.31345504860765789763478467618, 11.16580228767077314025732420675

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