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Properties

Label 2-1352-13.4-c1-0-10
Degree $2$
Conductor $1352$
Sign $-0.0771 - 0.997i$
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 − 0.561i·5-s + (2.21 − 1.28i)7-s + (−1.78 − 3.08i)9-s + (−4.43 − 2.56i)11-s + (1.24 + 0.719i)15-s + (2.84 + 4.92i)17-s + (4.43 − 2.56i)19-s + 6.56i·21-s + (−4 + 6.92i)23-s + 4.68·25-s + 1.43·27-s + (1 − 1.73i)29-s + 4i·31-s + (11.3 − 6.56i)33-s + ⋯
L(s)  = 1  + (−0.739 + 1.28i)3-s − 0.251i·5-s + (0.838 − 0.484i)7-s + (−0.593 − 1.02i)9-s + (−1.33 − 0.772i)11-s + (0.321 + 0.185i)15-s + (0.689 + 1.19i)17-s + (1.01 − 0.587i)19-s + 1.43i·21-s + (−0.834 + 1.44i)23-s + 0.936·25-s + 0.276·27-s + (0.185 − 0.321i)29-s + 0.718i·31-s + (1.97 − 1.14i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0771 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0771 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $-0.0771 - 0.997i$
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: \((2,\ 1352,\ (\ :1/2),\ -0.0771 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.202698967\)
\(L(\frac12)\) \(\approx\) \(1.202698967\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.561iT - 5T^{2} \)
7 \( 1 + (-2.21 + 1.28i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.43 + 2.56i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.84 - 4.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.43 + 2.56i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1 + 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + (-8.38 - 4.84i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.70 - 1.56i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.71 - 4.70i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.315iT - 47T^{2} \)
53 \( 1 - 3.12T + 53T^{2} \)
59 \( 1 + (4.43 - 2.56i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.56 + 9.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.43 - 2.56i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.65 - 3.84i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 2.24iT - 83T^{2} \)
89 \( 1 + (-8.66 - 5i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.14 - 4.12i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.991788201733549593096802658119, −9.225031597332471402478890628469, −8.059397480504980650710425158513, −7.71300654571312154743477924262, −6.17261333743161442365623440663, −5.38963059827585632046480432206, −4.89570008770867140134468729061, −3.97680483379128826602630211616, −2.99252871270861561782677401895, −1.13739928461083132435550556852, 0.64541948068779376037805825399, 2.01019100074603563484998375415, 2.78542316297166651237670568866, 4.58951599214494589023023710366, 5.38762179396391275054469431007, 6.01390849983842934441205526879, 7.15324025852364810179516494398, 7.59909204427250358947990642213, 8.213198921444347537261654251503, 9.425556771531291170896099830862

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