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Properties

Label 2-1352-13.4-c1-0-0
Degree $2$
Conductor $1352$
Sign $-0.967 + 0.252i$
Analytic cond. $10.7957$
Root an. cond. $3.28569$
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  + (−1.13 + 1.97i)3-s − 1.12i·5-s + (2.26 − 1.30i)7-s + (−1.09 − 1.89i)9-s + (−2.26 − 1.30i)11-s + (2.21 + 1.27i)15-s + (−2.77 − 4.81i)17-s + (−6.88 + 3.97i)19-s + 5.94i·21-s + (−1.13 + 1.97i)23-s + 3.74·25-s − 1.85·27-s + (−3.89 + 6.75i)29-s + 8.26i·31-s + (5.14 − 2.97i)33-s + ⋯
L(s)  = 1  + (−0.657 + 1.13i)3-s − 0.501i·5-s + (0.854 − 0.493i)7-s + (−0.364 − 0.631i)9-s + (−0.681 − 0.393i)11-s + (0.571 + 0.329i)15-s + (−0.673 − 1.16i)17-s + (−1.57 + 0.911i)19-s + 1.29i·21-s + (−0.237 + 0.411i)23-s + 0.748·25-s − 0.356·27-s + (−0.724 + 1.25i)29-s + 1.48i·31-s + (0.896 − 0.517i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $-0.967 + 0.252i$
Analytic conductor: \(10.7957\)
Root analytic conductor: \(3.28569\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1352} (1161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1352,\ (\ :1/2),\ -0.967 + 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2123076434\)
\(L(\frac12)\) \(\approx\) \(0.2123076434\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (1.13 - 1.97i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.12iT - 5T^{2} \)
7 \( 1 + (-2.26 + 1.30i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.26 + 1.30i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.77 + 4.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.88 - 3.97i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.13 - 1.97i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.89 - 6.75i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.26iT - 31T^{2} \)
37 \( 1 + (3.11 + 1.80i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.96 + 2.86i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.01 - 3.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.66iT - 47T^{2} \)
53 \( 1 + 9.54T + 53T^{2} \)
59 \( 1 + (3.31 - 1.91i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.84 + 6.65i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.52 - 0.881i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.880 - 0.508i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 0.323iT - 73T^{2} \)
79 \( 1 + 2.66T + 79T^{2} \)
83 \( 1 + 2.77iT - 83T^{2} \)
89 \( 1 + (-9.99 - 5.77i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.5 - 7.82i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.25252524108270070593565205688, −9.245435699872126653965712479828, −8.581451452452803378992068613884, −7.71891385407968536743759234249, −6.69622886044804224882740197740, −5.53573389085613400888930451565, −4.90345108717356295915100288908, −4.39618436469124154574358754581, −3.27887828600260924939225756962, −1.66727260574101310899267056837, 0.089431085805345631277611551993, 1.85652968182424199270854521903, 2.41067965872056017977528390456, 4.13545137171559726521454126583, 5.06342432930428046863991386898, 6.16223720039183958559248133468, 6.52848751108282381195237465053, 7.53442374009099168993483866666, 8.162808094961643459878578572117, 8.954170796500351693710078112277

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