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Properties

Label 2-1450-145.12-c1-0-13
Degree $2$
Conductor $1450$
Sign $0.191 + 0.981i$
Analytic cond. $11.5783$
Root an. cond. $3.40269$
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  − i·2-s − 2.92·3-s − 4-s + 2.92i·6-s + (−0.915 − 0.915i)7-s + i·8-s + 5.53·9-s + (−2.84 − 2.84i)11-s + 2.92·12-s + (2.39 + 2.39i)13-s + (−0.915 + 0.915i)14-s + 16-s + 0.819i·17-s − 5.53i·18-s + (−3.43 + 3.43i)19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.68·3-s − 0.5·4-s + 1.19i·6-s + (−0.345 − 0.345i)7-s + 0.353i·8-s + 1.84·9-s + (−0.858 − 0.858i)11-s + 0.843·12-s + (0.664 + 0.664i)13-s + (−0.244 + 0.244i)14-s + 0.250·16-s + 0.198i·17-s − 1.30i·18-s + (−0.787 + 0.787i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.191 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1450\)    =    \(2 \cdot 5^{2} \cdot 29\)
Sign: $0.191 + 0.981i$
Analytic conductor: \(11.5783\)
Root analytic conductor: \(3.40269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1450} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1450,\ (\ :1/2),\ 0.191 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5924455782\)
\(L(\frac12)\) \(\approx\) \(0.5924455782\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 \)
29 \( 1 + (-4.57 + 2.84i)T \)
good3 \( 1 + 2.92T + 3T^{2} \)
7 \( 1 + (0.915 + 0.915i)T + 7iT^{2} \)
11 \( 1 + (2.84 + 2.84i)T + 11iT^{2} \)
13 \( 1 + (-2.39 - 2.39i)T + 13iT^{2} \)
17 \( 1 - 0.819iT - 17T^{2} \)
19 \( 1 + (3.43 - 3.43i)T - 19iT^{2} \)
23 \( 1 + (5.47 - 5.47i)T - 23iT^{2} \)
31 \( 1 + (-5.31 - 5.31i)T + 31iT^{2} \)
37 \( 1 - 1.12T + 37T^{2} \)
41 \( 1 + (1.30 - 1.30i)T - 41iT^{2} \)
43 \( 1 - 9.87T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 + (-5.31 + 5.31i)T - 53iT^{2} \)
59 \( 1 + 1.52iT - 59T^{2} \)
61 \( 1 + (5.47 + 5.47i)T + 61iT^{2} \)
67 \( 1 + (-6.97 + 6.97i)T - 67iT^{2} \)
71 \( 1 + 9.83iT - 71T^{2} \)
73 \( 1 - 1.21iT - 73T^{2} \)
79 \( 1 + (3.27 - 3.27i)T - 79iT^{2} \)
83 \( 1 + (-1 + i)T - 83iT^{2} \)
89 \( 1 + (-4.60 + 4.60i)T - 89iT^{2} \)
97 \( 1 + 8.97T + 97T^{2} \)
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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

−9.851508141551950348201811762167, −8.574248885269229118693148354614, −7.78204881967009928947290568896, −6.47535311579790981465946097457, −6.08705283912337563901380125018, −5.18660338449901573136200978203, −4.28905294629226155722009053362, −3.39892318838290079482127261889, −1.77615933517696420499915163070, −0.51730294771622726410220453483, 0.68473916334256611110191689870, 2.54983279897889415560185378282, 4.29684066014350716815867402804, 4.82405773868707530916576093063, 5.81140348818304765627362772014, 6.23275448012264078584453371137, 7.01601498377212080420473448827, 7.909048037138377125045583956231, 8.800108881813118064424084249825, 9.996303516827491037862504169269

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