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Properties

Label 2-20e2-25.19-c1-0-8
Degree $2$
Conductor $400$
Sign $0.277 + 0.960i$
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  + (−0.0350 − 0.0481i)3-s + (−0.243 − 2.22i)5-s + 1.71i·7-s + (0.925 − 2.84i)9-s + (−0.573 − 1.76i)11-s + (0.533 + 0.173i)13-s + (−0.0985 + 0.0895i)15-s + (2.29 − 3.15i)17-s + (−4.69 − 3.40i)19-s + (0.0827 − 0.0601i)21-s + (6.95 − 2.25i)23-s + (−4.88 + 1.08i)25-s + (−0.339 + 0.110i)27-s + (2.13 − 1.55i)29-s + (−1.56 − 1.13i)31-s + ⋯
L(s)  = 1  + (−0.0202 − 0.0278i)3-s + (−0.108 − 0.994i)5-s + 0.648i·7-s + (0.308 − 0.949i)9-s + (−0.172 − 0.532i)11-s + (0.148 + 0.0481i)13-s + (−0.0254 + 0.0231i)15-s + (0.556 − 0.766i)17-s + (−1.07 − 0.782i)19-s + (0.0180 − 0.0131i)21-s + (1.45 − 0.471i)23-s + (−0.976 + 0.216i)25-s + (−0.0653 + 0.0212i)27-s + (0.396 − 0.288i)29-s + (−0.280 − 0.203i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.277 + 0.960i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.277 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01369 - 0.762288i\)
\(L(\frac12)\) \(\approx\) \(1.01369 - 0.762288i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.243 + 2.22i)T \)
good3 \( 1 + (0.0350 + 0.0481i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 - 1.71iT - 7T^{2} \)
11 \( 1 + (0.573 + 1.76i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.533 - 0.173i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.29 + 3.15i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (4.69 + 3.40i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-6.95 + 2.25i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-2.13 + 1.55i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.56 + 1.13i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.725 - 0.235i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.00 - 9.25i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 9.68iT - 43T^{2} \)
47 \( 1 + (-6.11 - 8.42i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.654 - 0.900i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.56 + 4.81i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.18 - 12.8i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (0.733 - 1.01i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (10.8 - 7.91i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.88 - 1.26i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.25 - 3.81i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.536 - 0.738i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (1.61 + 4.97i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-8.54 - 11.7i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.29046468864710405887082976307, −10.05746227182764487996910221565, −8.920495841606337698976053128871, −8.753183490240656630659839295270, −7.35544729636807461038544397349, −6.24075364652821014609966711322, −5.24414941200462198029986453941, −4.20838973725144079613350811569, −2.78059611543843484039724790666, −0.888189619125443349592009144819, 1.91240792561667663957398570312, 3.38224067593738820637766392539, 4.47804220377239275462041105104, 5.76618248390494914030577634694, 6.97469158315510959053635772195, 7.55508523254381904472272384624, 8.578850358210343406171883612569, 10.05493452682998733994648874932, 10.50964311990614001241461335567, 11.15782624250177507613279662292

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