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Properties

Label 2-1352-13.4-c1-0-14
Degree $2$
Conductor $1352$
Sign $-0.252 - 0.967i$
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  + (−0.5 + 0.866i)3-s + 2i·5-s + (0.866 − 0.5i)7-s + (1 + 1.73i)9-s + (0.866 + 0.5i)11-s + (−1.73 − i)15-s + (1.5 + 2.59i)17-s + (6.06 − 3.5i)19-s + 0.999i·21-s + (0.5 − 0.866i)23-s + 25-s − 5·27-s + (−1.5 + 2.59i)29-s + 8i·31-s + (−0.866 + 0.499i)33-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + 0.894i·5-s + (0.327 − 0.188i)7-s + (0.333 + 0.577i)9-s + (0.261 + 0.150i)11-s + (−0.447 − 0.258i)15-s + (0.363 + 0.630i)17-s + (1.39 − 0.802i)19-s + 0.218i·21-s + (0.104 − 0.180i)23-s + 0.200·25-s − 0.962·27-s + (−0.278 + 0.482i)29-s + 1.43i·31-s + (−0.150 + 0.0870i)33-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $-0.252 - 0.967i$
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: \((2,\ 1352,\ (\ :1/2),\ -0.252 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.589799447\)
\(L(\frac12)\) \(\approx\) \(1.589799447\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 2iT - 5T^{2} \)
7 \( 1 + (-0.866 + 0.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.06 + 3.5i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8iT - 31T^{2} \)
37 \( 1 + (0.866 + 0.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (9.52 + 5.5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.5 - 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 12iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-7.79 + 4.5i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.5 - 7.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.59 - 1.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.33 - 2.5i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + (0.866 + 0.5i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.866 - 0.5i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.08557643998578090765883378059, −9.149998095635525488490808974261, −8.177906693895764028331741143682, −7.22392171219500948977376478648, −6.77433542292604375782681084841, −5.48887303661766617888136386845, −4.87496797952156655339496346753, −3.78314445349434116166972296950, −2.87163952146732100740331646262, −1.48709679991667142071277945232, 0.75420910160765893378513946498, 1.68900415258755780162544514081, 3.24840268788942516484792181657, 4.29695895269793172487369087327, 5.30683053226295613866815071170, 5.93005180789347523160681750814, 6.99245783475292735309003833647, 7.73727034077698753506853521557, 8.512738166400534325173805601025, 9.512379734533043134521083561143

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