↑

Properties

Label 2-20e2-25.14-c1-0-1
Degree $2$
Conductor $400$
Sign $-0.636 - 0.771i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.07 − 0.350i)3-s + (0.279 + 2.21i)5-s + 0.768i·7-s + (−1.38 − 1.00i)9-s + (−2.47 + 1.79i)11-s + (0.132 − 0.182i)13-s + (0.476 − 2.48i)15-s + (−5.57 + 1.81i)17-s + (1.90 + 5.86i)19-s + (0.269 − 0.828i)21-s + (−3.05 − 4.19i)23-s + (−4.84 + 1.23i)25-s + (3.14 + 4.32i)27-s + (−0.0146 + 0.0451i)29-s + (1.80 + 5.55i)31-s + ⋯
L(s)  = 1  + (−0.622 − 0.202i)3-s + (0.124 + 0.992i)5-s + 0.290i·7-s + (−0.462 − 0.335i)9-s + (−0.746 + 0.542i)11-s + (0.0368 − 0.0506i)13-s + (0.122 − 0.642i)15-s + (−1.35 + 0.439i)17-s + (0.437 + 1.34i)19-s + (0.0587 − 0.180i)21-s + (−0.636 − 0.875i)23-s + (−0.968 + 0.247i)25-s + (0.604 + 0.832i)27-s + (−0.00272 + 0.00838i)29-s + (0.324 + 0.997i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.636 - 0.771i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.636 - 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.256344 + 0.544051i\)
\(L(\frac12)\) \(\approx\) \(0.256344 + 0.544051i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.279 - 2.21i)T \)
good3 \( 1 + (1.07 + 0.350i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 - 0.768iT - 7T^{2} \)
11 \( 1 + (2.47 - 1.79i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.132 + 0.182i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.57 - 1.81i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.90 - 5.86i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (3.05 + 4.19i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.0146 - 0.0451i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.80 - 5.55i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.99 - 6.86i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-5.71 - 4.14i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 3.14iT - 43T^{2} \)
47 \( 1 + (10.8 + 3.51i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.01 - 0.980i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.36 - 1.71i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-6.43 + 4.67i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (0.281 - 0.0914i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-4.13 + 12.7i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (8.49 + 11.6i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.40 + 4.32i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-6.45 + 2.09i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (6.90 - 5.01i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-18.2 - 5.92i)T + (78.4 + 57.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.64182051353849509479826955622, −10.61036230077864694700727438998, −10.14366724069766540554903697329, −8.815065147861026818022470269521, −7.81700503152159962124696768276, −6.64581264524711927023937244793, −6.10449959824262339591938358023, −4.94390337920576725004568514662, −3.42940876702459651115542590497, −2.14943042546359144284782786897, 0.39895891690679981505540036055, 2.45224089663190267540979726274, 4.22234064527458489386971771095, 5.16591649889002759551745113904, 5.86361892721398742146884171583, 7.20607482739105395332828550943, 8.305196942527670618327147311347, 9.089685633753173343502273241659, 10.07353306556917509279389511130, 11.27184778596178173358247413434

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