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Properties

Label 2-20e2-80.37-c2-0-65
Degree $2$
Conductor $400$
Sign $-0.685 + 0.728i$
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  + (1.99 − 0.0957i)2-s − 4.94i·3-s + (3.98 − 0.382i)4-s + (−0.473 − 9.87i)6-s + (−3.22 + 3.22i)7-s + (7.91 − 1.14i)8-s − 15.4·9-s + (−7.67 − 7.67i)11-s + (−1.89 − 19.6i)12-s − 22.2i·13-s + (−6.13 + 6.75i)14-s + (15.7 − 3.04i)16-s + (3.76 + 3.76i)17-s + (−30.8 + 1.47i)18-s + (−0.809 − 0.809i)19-s + ⋯
L(s)  = 1  + (0.998 − 0.0478i)2-s − 1.64i·3-s + (0.995 − 0.0956i)4-s + (−0.0789 − 1.64i)6-s + (−0.460 + 0.460i)7-s + (0.989 − 0.143i)8-s − 1.71·9-s + (−0.697 − 0.697i)11-s + (−0.157 − 1.64i)12-s − 1.70i·13-s + (−0.438 + 0.482i)14-s + (0.981 − 0.190i)16-s + (0.221 + 0.221i)17-s + (−1.71 + 0.0821i)18-s + (−0.0426 − 0.0426i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 + 0.728i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.685 + 0.728i$
Analytic conductor: \(10.8992\)
Root analytic conductor: \(3.30139\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1),\ -0.685 + 0.728i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.10283 - 2.55094i\)
\(L(\frac12)\) \(\approx\) \(1.10283 - 2.55094i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.99 + 0.0957i)T \)
5 \( 1 \)
good3 \( 1 + 4.94iT - 9T^{2} \)
7 \( 1 + (3.22 - 3.22i)T - 49iT^{2} \)
11 \( 1 + (7.67 + 7.67i)T + 121iT^{2} \)
13 \( 1 + 22.2iT - 169T^{2} \)
17 \( 1 + (-3.76 - 3.76i)T + 289iT^{2} \)
19 \( 1 + (0.809 + 0.809i)T + 361iT^{2} \)
23 \( 1 + (12.2 + 12.2i)T + 529iT^{2} \)
29 \( 1 + (-27.1 - 27.1i)T + 841iT^{2} \)
31 \( 1 - 25.5T + 961T^{2} \)
37 \( 1 + 8.62iT - 1.36e3T^{2} \)
41 \( 1 - 73.4iT - 1.68e3T^{2} \)
43 \( 1 - 17.5T + 1.84e3T^{2} \)
47 \( 1 + (17.4 + 17.4i)T + 2.20e3iT^{2} \)
53 \( 1 - 29.3T + 2.80e3T^{2} \)
59 \( 1 + (-72.6 + 72.6i)T - 3.48e3iT^{2} \)
61 \( 1 + (-19.7 + 19.7i)T - 3.72e3iT^{2} \)
67 \( 1 + 14.0T + 4.48e3T^{2} \)
71 \( 1 + 91.3iT - 5.04e3T^{2} \)
73 \( 1 + (-42.7 - 42.7i)T + 5.32e3iT^{2} \)
79 \( 1 - 46.7iT - 6.24e3T^{2} \)
83 \( 1 - 82.9iT - 6.88e3T^{2} \)
89 \( 1 - 131.T + 7.92e3T^{2} \)
97 \( 1 + (-30.5 - 30.5i)T + 9.40e3iT^{2} \)
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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

−11.03497332583844078093994984431, −10.14061779014689518003399392398, −8.273122045835437759946858178124, −7.890013522208455669062267155695, −6.66899341345552579155860135863, −6.01684660967794430961246144987, −5.17997002585986271238479522207, −3.19739291262841963215331215548, −2.49158205772522508444323970854, −0.870038009883744893538828545675, 2.42824196935769755372209981510, 3.78919979831426990867955966955, 4.36963569515733323665704258479, 5.24862757855647822589226847353, 6.41513368479514926207097652523, 7.47353509750978975760674742394, 8.874435479166872282035617398276, 10.01889456501125549643659949149, 10.29518360222993289744280322858, 11.48473751820691497229522616657

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