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Properties

Label 2-20e2-16.11-c2-0-45
Degree $2$
Conductor $400$
Sign $0.357 + 0.933i$
Analytic cond. $10.8992$
Root an. cond. $3.30139$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.403 − 1.95i)2-s + (1.45 + 1.45i)3-s + (−3.67 − 1.58i)4-s + (3.44 − 2.26i)6-s + 11.3·7-s + (−4.57 + 6.55i)8-s − 4.75i·9-s + (−1.50 + 1.50i)11-s + (−3.05 − 7.66i)12-s + (0.454 − 0.454i)13-s + (4.59 − 22.3i)14-s + (11.0 + 11.6i)16-s + 1.99·17-s + (−9.30 − 1.91i)18-s + (5.07 + 5.07i)19-s + ⋯
L(s)  = 1  + (0.201 − 0.979i)2-s + (0.485 + 0.485i)3-s + (−0.918 − 0.395i)4-s + (0.573 − 0.377i)6-s + 1.62·7-s + (−0.572 + 0.819i)8-s − 0.527i·9-s + (−0.137 + 0.137i)11-s + (−0.254 − 0.638i)12-s + (0.0349 − 0.0349i)13-s + (0.328 − 1.59i)14-s + (0.687 + 0.726i)16-s + 0.117·17-s + (−0.516 − 0.106i)18-s + (0.267 + 0.267i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 + 0.933i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.357 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.357 + 0.933i$
Analytic conductor: \(10.8992\)
Root analytic conductor: \(3.30139\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1),\ 0.357 + 0.933i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.95562 - 1.34541i\)
\(L(\frac12)\) \(\approx\) \(1.95562 - 1.34541i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.403 + 1.95i)T \)
5 \( 1 \)
good3 \( 1 + (-1.45 - 1.45i)T + 9iT^{2} \)
7 \( 1 - 11.3T + 49T^{2} \)
11 \( 1 + (1.50 - 1.50i)T - 121iT^{2} \)
13 \( 1 + (-0.454 + 0.454i)T - 169iT^{2} \)
17 \( 1 - 1.99T + 289T^{2} \)
19 \( 1 + (-5.07 - 5.07i)T + 361iT^{2} \)
23 \( 1 - 41.9T + 529T^{2} \)
29 \( 1 + (-7.01 + 7.01i)T - 841iT^{2} \)
31 \( 1 + 33.3iT - 961T^{2} \)
37 \( 1 + (44.5 + 44.5i)T + 1.36e3iT^{2} \)
41 \( 1 + 51.6iT - 1.68e3T^{2} \)
43 \( 1 + (37.7 - 37.7i)T - 1.84e3iT^{2} \)
47 \( 1 - 16.2iT - 2.20e3T^{2} \)
53 \( 1 + (-67.7 - 67.7i)T + 2.80e3iT^{2} \)
59 \( 1 + (-34.2 + 34.2i)T - 3.48e3iT^{2} \)
61 \( 1 + (67.1 - 67.1i)T - 3.72e3iT^{2} \)
67 \( 1 + (-9.87 - 9.87i)T + 4.48e3iT^{2} \)
71 \( 1 + 74.7T + 5.04e3T^{2} \)
73 \( 1 - 101. iT - 5.32e3T^{2} \)
79 \( 1 - 63.6iT - 6.24e3T^{2} \)
83 \( 1 + (57.1 + 57.1i)T + 6.88e3iT^{2} \)
89 \( 1 - 33.6iT - 7.92e3T^{2} \)
97 \( 1 - 98.2T + 9.40e3T^{2} \)
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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

−10.94095777604767040555748397603, −10.12655244622927551687293491446, −9.067748292301371917252030453143, −8.533367342618120159248889316979, −7.38488383129738569929539744275, −5.62480006304189358936686095227, −4.71339778298321764991809620257, −3.81159280361036151084952034681, −2.54871321175922551219591855040, −1.16346992374726753168171873697, 1.44221066652975629548065956019, 3.13348645308430082168031531980, 4.85380217976362940507219674801, 5.16286509942558098538790702962, 6.80961389996111518237784282464, 7.49789561380181077506517002887, 8.421368220721358476485469476720, 8.769944010031680854894213307517, 10.30290396513637851076854830895, 11.30066826159815276621404686892

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