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Properties

Label 2-1352-13.3-c1-0-14
Degree $2$
Conductor $1352$
Sign $0.859 + 0.511i$
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 + 3.56·5-s + (−1.28 + 2.21i)7-s + (−1.78 + 3.08i)9-s + (1.28 + 2.21i)11-s + (−4.56 − 7.90i)15-s + (2.5 − 4.33i)17-s + (−1.28 + 2.21i)19-s + 6.56·21-s + (1.84 + 3.19i)23-s + 7.68·25-s + 1.43·27-s + (2.5 + 4.33i)29-s + 8·31-s + (3.28 − 5.68i)33-s + ⋯
L(s)  = 1  + (−0.739 − 1.28i)3-s + 1.59·5-s + (−0.484 + 0.838i)7-s + (−0.593 + 1.02i)9-s + (0.386 + 0.668i)11-s + (−1.17 − 2.03i)15-s + (0.606 − 1.05i)17-s + (−0.293 + 0.508i)19-s + 1.43·21-s + (0.384 + 0.665i)23-s + 1.53·25-s + 0.276·27-s + (0.464 + 0.804i)29-s + 1.43·31-s + (0.571 − 0.989i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(10.7957\)
Root analytic conductor: \(3.28569\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1352} (1329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1352,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.713675536\)
\(L(\frac12)\) \(\approx\) \(1.713675536\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.56T + 5T^{2} \)
7 \( 1 + (1.28 - 2.21i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.28 - 2.21i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.28 - 2.21i)T + (-9.5 - 16.4i)T^{2} \)
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 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.62 - 8.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.28 + 5.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 - 4.43T + 53T^{2} \)
59 \( 1 + (1.28 - 2.21i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.62 + 6.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.71 + 8.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.84 + 6.65i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.31T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 2.24T + 83T^{2} \)
89 \( 1 + (4.84 + 8.38i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.40 + 2.43i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.590121012115573191003676450480, −8.875583518020704590566407266684, −7.68277887032135208539524743433, −6.82038228051893563745711433232, −6.23869051461533920903594270592, −5.65947962107202692123148658640, −4.88234611187480801511338083612, −2.95013157785198993022434268112, −2.05100058898350622996282160770, −1.14412205837935814565843796891, 0.962992648489410215951832735586, 2.61838802922449545594909076497, 3.82778874519674465465160245302, 4.59122893353911150005252863438, 5.62370532111394074763288321203, 6.12240810852411235831547752264, 6.83888003823779803482294646580, 8.335475821820926640800038246170, 9.198874069745482091914801792556, 9.933254544912610104829337063502

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