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Properties

Label 2-1352-13.10-c1-0-37
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  + (−1.28 − 2.21i)3-s − 3.56i·5-s + (−2.21 − 1.28i)7-s + (−1.78 + 3.08i)9-s + (−2.21 + 1.28i)11-s + (−7.90 + 4.56i)15-s + (−2.5 + 4.33i)17-s + (2.21 + 1.28i)19-s + 6.56i·21-s + (−1.84 − 3.19i)23-s − 7.68·25-s + 1.43·27-s + (2.5 + 4.33i)29-s − 8i·31-s + (5.68 + 3.28i)33-s + ⋯
L(s)  = 1  + (−0.739 − 1.28i)3-s − 1.59i·5-s + (−0.838 − 0.484i)7-s + (−0.593 + 1.02i)9-s + (−0.668 + 0.386i)11-s + (−2.03 + 1.17i)15-s + (−0.606 + 1.05i)17-s + (0.508 + 0.293i)19-s + 1.43i·21-s + (−0.384 − 0.665i)23-s − 1.53·25-s + 0.276·27-s + (0.464 + 0.804i)29-s − 1.43i·31-s + (0.989 + 0.571i)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: $\chi_{1352} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1352,\ (\ :1/2),\ 0.252 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2254628174\)
\(L(\frac12)\) \(\approx\) \(0.2254628174\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.28 + 2.21i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 3.56iT - 5T^{2} \)
7 \( 1 + (2.21 + 1.28i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.21 - 1.28i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.21 - 1.28i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.84 + 3.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8iT - 31T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-8.00 + 4.62i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.28 - 5.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 - 4.43T + 53T^{2} \)
59 \( 1 + (2.21 + 1.28i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.62 + 6.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.17 - 4.71i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.65 + 3.84i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.31iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 2.24iT - 83T^{2} \)
89 \( 1 + (-8.38 + 4.84i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.43 + 1.40i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.830922338437435575912480408624, −8.026332247874949491495064369999, −7.36549294874960361172972768265, −6.38959427862682783812987744329, −5.79084398110462558147727903935, −4.85931046285219653705180319813, −3.91386117800414127998395683228, −2.21410168098523973987471451964, −1.12175117746517501257456016481, −0.11311401418781187692683725998, 2.70679949368567254106977845223, 3.16747372757871725369622688526, 4.28127947334557937436318880837, 5.34054460216394905028266135705, 6.03157232132997939385868546169, 6.80076080752900102322595249755, 7.64109814708583016989486841491, 8.982064479955757576568796334808, 9.715248456614793907058923309899, 10.24827248424488181268995019485

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