↑

Properties

Label 2-20e2-100.23-c1-0-7
Degree $2$
Conductor $400$
Sign $0.866 - 0.499i$
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  + (0.630 + 1.23i)3-s + (2.13 − 0.668i)5-s + (1.04 + 1.04i)7-s + (0.630 − 0.868i)9-s + (0.0991 + 0.136i)11-s + (−0.0572 − 0.361i)13-s + (2.17 + 2.21i)15-s + (−1.20 − 0.614i)17-s + (−0.488 − 1.50i)19-s + (−0.635 + 1.95i)21-s + (−1.02 + 6.45i)23-s + (4.10 − 2.85i)25-s + (5.58 + 0.884i)27-s + (−3.38 − 1.09i)29-s + (−3.16 + 1.02i)31-s + ⋯
L(s)  = 1  + (0.363 + 0.714i)3-s + (0.954 − 0.299i)5-s + (0.396 + 0.396i)7-s + (0.210 − 0.289i)9-s + (0.0298 + 0.0411i)11-s + (−0.0158 − 0.100i)13-s + (0.560 + 0.572i)15-s + (−0.292 − 0.149i)17-s + (−0.111 − 0.344i)19-s + (−0.138 + 0.426i)21-s + (−0.213 + 1.34i)23-s + (0.821 − 0.570i)25-s + (1.07 + 0.170i)27-s + (−0.628 − 0.204i)29-s + (−0.568 + 0.184i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.866 - 0.499i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.866 - 0.499i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79449 + 0.480816i\)
\(L(\frac12)\) \(\approx\) \(1.79449 + 0.480816i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.13 + 0.668i)T \)
good3 \( 1 + (-0.630 - 1.23i)T + (-1.76 + 2.42i)T^{2} \)
7 \( 1 + (-1.04 - 1.04i)T + 7iT^{2} \)
11 \( 1 + (-0.0991 - 0.136i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.0572 + 0.361i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (1.20 + 0.614i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (0.488 + 1.50i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.02 - 6.45i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (3.38 + 1.09i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.16 - 1.02i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.57 - 0.566i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-6.74 - 4.89i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (4.31 - 4.31i)T - 43iT^{2} \)
47 \( 1 + (-3.72 + 1.89i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (2.34 - 1.19i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (10.3 + 7.52i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (7.31 - 5.31i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-5.12 + 10.0i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (15.4 + 5.02i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-14.1 - 2.23i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-4.87 + 15.0i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (11.4 + 5.85i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (-2.77 - 3.82i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (5.33 + 10.4i)T + (-57.0 + 78.4i)T^{2} \)
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

−11.21709205941158509554967537478, −10.26685439659550602823025785497, −9.366445516792286946047113937939, −9.016898966948261751045832157411, −7.77024350932027622070194114932, −6.48590241228071902584790262799, −5.43938007969412039663941508153, −4.52422871256670785070023720123, −3.20271363353397448133595388599, −1.74419897596815614943915923823, 1.57462473495733115237383554088, 2.58515390217518739582219812714, 4.25303316386940058731818553301, 5.53040432151865132463226515862, 6.62062030335728336139341648538, 7.39150950313042174763394493379, 8.374301057967284906793382020856, 9.326360916534213510807885887682, 10.45820391118971198762482964729, 10.93035183469784394363948451119

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