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Properties

Label 2-20e2-16.13-c1-0-6
Degree $2$
Conductor $400$
Sign $0.577 - 0.816i$
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  + (−0.719 − 1.21i)2-s + (1.66 + 1.66i)3-s + (−0.965 + 1.75i)4-s + (0.831 − 3.23i)6-s + 1.87i·7-s + (2.82 − 0.0835i)8-s + 2.56i·9-s + (−3.29 + 3.29i)11-s + (−4.53 + 1.31i)12-s + (1.90 + 1.90i)13-s + (2.28 − 1.34i)14-s + (−2.13 − 3.38i)16-s + 2.57·17-s + (3.12 − 1.84i)18-s + (−5.76 − 5.76i)19-s + ⋯
L(s)  = 1  + (−0.508 − 0.861i)2-s + (0.962 + 0.962i)3-s + (−0.482 + 0.875i)4-s + (0.339 − 1.31i)6-s + 0.708i·7-s + (0.999 − 0.0295i)8-s + 0.854i·9-s + (−0.994 + 0.994i)11-s + (−1.30 + 0.378i)12-s + (0.527 + 0.527i)13-s + (0.609 − 0.360i)14-s + (−0.533 − 0.845i)16-s + 0.623·17-s + (0.735 − 0.434i)18-s + (−1.32 − 1.32i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13237 + 0.586442i\)
\(L(\frac12)\) \(\approx\) \(1.13237 + 0.586442i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.719 + 1.21i)T \)
5 \( 1 \)
good3 \( 1 + (-1.66 - 1.66i)T + 3iT^{2} \)
7 \( 1 - 1.87iT - 7T^{2} \)
11 \( 1 + (3.29 - 3.29i)T - 11iT^{2} \)
13 \( 1 + (-1.90 - 1.90i)T + 13iT^{2} \)
17 \( 1 - 2.57T + 17T^{2} \)
19 \( 1 + (5.76 + 5.76i)T + 19iT^{2} \)
23 \( 1 - 7.58iT - 23T^{2} \)
29 \( 1 + (-6.45 - 6.45i)T + 29iT^{2} \)
31 \( 1 + 0.799T + 31T^{2} \)
37 \( 1 + (-2.69 + 2.69i)T - 37iT^{2} \)
41 \( 1 + 0.946iT - 41T^{2} \)
43 \( 1 + (-0.829 + 0.829i)T - 43iT^{2} \)
47 \( 1 - 1.52T + 47T^{2} \)
53 \( 1 + (-6.97 + 6.97i)T - 53iT^{2} \)
59 \( 1 + (-6.84 + 6.84i)T - 59iT^{2} \)
61 \( 1 + (6.87 + 6.87i)T + 61iT^{2} \)
67 \( 1 + (3.73 + 3.73i)T + 67iT^{2} \)
71 \( 1 - 9.34iT - 71T^{2} \)
73 \( 1 - 0.886iT - 73T^{2} \)
79 \( 1 - 3.07T + 79T^{2} \)
83 \( 1 + (0.989 + 0.989i)T + 83iT^{2} \)
89 \( 1 + 10.0iT - 89T^{2} \)
97 \( 1 + 7.16T + 97T^{2} \)
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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

−11.14955401905817212893050207862, −10.34229802019272476031645310572, −9.563846100189349258638637902219, −8.901554687059766008335262495414, −8.225544639531834029011331437251, −7.06487366914386396129262538497, −5.15824948665382266050123661469, −4.20395781599019434675546987448, −3.07617833886177928597087530960, −2.13029458997342918225257678630, 0.931343818864742879498469427387, 2.61174475725217469625426272724, 4.18423146423356621436752892492, 5.77981407672026126791065878085, 6.56908306785705737488039694102, 7.77015730154564548486989255501, 8.139158076767083732262238336728, 8.770164964541090494410765624554, 10.37296220879516723088065899845, 10.55593539542168637091797729917

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