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Properties

Label 2-20e2-100.3-c1-0-10
Degree $2$
Conductor $400$
Sign $0.998 + 0.0627i$
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  + (2.13 − 0.337i)3-s + 2.23·5-s + (1.09 + 1.09i)7-s + (1.58 − 0.514i)9-s + (−1.97 − 0.642i)11-s + (0.587 − 1.15i)13-s + (4.77 − 0.755i)15-s + (−0.224 + 1.41i)17-s + (−6.56 + 4.77i)19-s + (2.70 + 1.96i)21-s + (−2.41 − 4.73i)23-s + 5.00·25-s + (−2.56 + 1.30i)27-s + (3.65 − 5.02i)29-s + (−3.63 − 4.99i)31-s + ⋯
L(s)  = 1  + (1.23 − 0.195i)3-s + 0.999·5-s + (0.413 + 0.413i)7-s + (0.528 − 0.171i)9-s + (−0.596 − 0.193i)11-s + (0.163 − 0.319i)13-s + (1.23 − 0.195i)15-s + (−0.0544 + 0.343i)17-s + (−1.50 + 1.09i)19-s + (0.589 + 0.428i)21-s + (−0.502 − 0.987i)23-s + 1.00·25-s + (−0.494 + 0.251i)27-s + (0.678 − 0.933i)29-s + (−0.651 − 0.897i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.998 + 0.0627i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.998 + 0.0627i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.25059 - 0.0707279i\)
\(L(\frac12)\) \(\approx\) \(2.25059 - 0.0707279i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - 2.23T \)
good3 \( 1 + (-2.13 + 0.337i)T + (2.85 - 0.927i)T^{2} \)
7 \( 1 + (-1.09 - 1.09i)T + 7iT^{2} \)
11 \( 1 + (1.97 + 0.642i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.587 + 1.15i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (0.224 - 1.41i)T + (-16.1 - 5.25i)T^{2} \)
19 \( 1 + (6.56 - 4.77i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (2.41 + 4.73i)T + (-13.5 + 18.6i)T^{2} \)
29 \( 1 + (-3.65 + 5.02i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.63 + 4.99i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.658 - 0.335i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (-2.49 - 7.67i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-2.49 + 2.49i)T - 43iT^{2} \)
47 \( 1 + (-1.64 - 10.4i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (1.37 + 8.68i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (1.75 + 5.39i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.25 - 3.85i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-4.77 - 0.755i)T + (63.7 + 20.7i)T^{2} \)
71 \( 1 + (-2.26 + 3.11i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (11.3 - 5.76i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (-12.6 - 9.15i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.0651 - 0.411i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (-0.613 - 0.199i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (17.7 - 2.81i)T + (92.2 - 29.9i)T^{2} \)
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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

−11.06667691689931479661136725446, −10.20650593992343495016602356671, −9.358211077390638663788489642307, −8.258283090907742119706654309360, −8.101480338464047980290631387490, −6.43993365252796959549664613249, −5.61983182915701890113819686894, −4.18076404950131563620312616694, −2.69145650578032974576761789299, −1.95896149325482798921002406567, 1.88956461838433175508318554582, 2.88769662913379546765654885299, 4.25398132803535968078688273914, 5.38395187307670113007781086198, 6.70857238927869990918941414359, 7.68499495386424051398203263168, 8.810415762451260256004993971946, 9.191511112864139571937451453078, 10.34026357640546063137194679491, 10.94306063886750871176068665100

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