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Properties

Label 2-20e2-80.29-c1-0-14
Degree $2$
Conductor $400$
Sign $0.755 + 0.655i$
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  + (−1 + i)2-s + (−1.15 + 1.15i)3-s − 2i·4-s − 2.31i·6-s − 4.31·7-s + (2 + 2i)8-s + 0.316i·9-s + (−0.158 + 0.158i)11-s + (2.31 + 2.31i)12-s + (2.31 − 2.31i)13-s + (4.31 − 4.31i)14-s − 4·16-s − 5.31i·17-s + (−0.316 − 0.316i)18-s + (−3.15 − 3.15i)19-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.668 + 0.668i)3-s − i·4-s − 0.945i·6-s − 1.63·7-s + (0.707 + 0.707i)8-s + 0.105i·9-s + (−0.0477 + 0.0477i)11-s + (0.668 + 0.668i)12-s + (0.642 − 0.642i)13-s + (1.15 − 1.15i)14-s − 16-s − 1.28i·17-s + (−0.0746 − 0.0746i)18-s + (−0.724 − 0.724i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.755 + 0.655i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.755 + 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.327628 - 0.122283i\)
\(L(\frac12)\) \(\approx\) \(0.327628 - 0.122283i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
5 \( 1 \)
good3 \( 1 + (1.15 - 1.15i)T - 3iT^{2} \)
7 \( 1 + 4.31T + 7T^{2} \)
11 \( 1 + (0.158 - 0.158i)T - 11iT^{2} \)
13 \( 1 + (-2.31 + 2.31i)T - 13iT^{2} \)
17 \( 1 + 5.31iT - 17T^{2} \)
19 \( 1 + (3.15 + 3.15i)T + 19iT^{2} \)
23 \( 1 - 6.31T + 23T^{2} \)
29 \( 1 + (2 + 2i)T + 29iT^{2} \)
31 \( 1 - 4.31T + 31T^{2} \)
37 \( 1 + (-7.31 - 7.31i)T + 37iT^{2} \)
41 \( 1 + 5iT - 41T^{2} \)
43 \( 1 + (5.63 + 5.63i)T + 43iT^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + (3.31 + 3.31i)T + 53iT^{2} \)
59 \( 1 + (5.31 - 5.31i)T - 59iT^{2} \)
61 \( 1 + (3.63 + 3.63i)T + 61iT^{2} \)
67 \( 1 + (5.84 - 5.84i)T - 67iT^{2} \)
71 \( 1 + 4.63iT - 71T^{2} \)
73 \( 1 + 13.3T + 73T^{2} \)
79 \( 1 - 2.31T + 79T^{2} \)
83 \( 1 + (3.84 - 3.84i)T - 83iT^{2} \)
89 \( 1 + 15.9iT - 89T^{2} \)
97 \( 1 - 6.63iT - 97T^{2} \)
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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

−10.83732332858155450967560691392, −10.12937891516479301623816038959, −9.462565303431754554209530847921, −8.581158302598023498668846905377, −7.25456688855058902186794362324, −6.45360145714697409758489680363, −5.57377127139635503344528923176, −4.58439430471250617386736743694, −2.91171466073016106397316905813, −0.33648628935244106308449450918, 1.34358882474634801164613284020, 3.05446969396274611960394349337, 4.08396030708408386228691820755, 6.19211826096264654229411762115, 6.50173859273743032882948429536, 7.68351179264771792381385171052, 8.892075678852223766343375893699, 9.560475734723259099821921327248, 10.57782604166109562755011579131, 11.29323990720140722126814347447

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