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Properties

Label 2-364-13.3-c1-0-0
Degree $2$
Conductor $364$
Sign $0.0128 - 0.999i$
Analytic cond. $2.90655$
Root an. cond. $1.70486$
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  + (−0.895 − 1.55i)3-s − 3.79·5-s + (−0.5 + 0.866i)7-s + (−0.104 + 0.180i)9-s + (1.89 + 3.28i)11-s + (2.5 + 2.59i)13-s + (3.39 + 5.88i)15-s + (−1.5 + 2.59i)17-s + (−3.68 + 6.38i)19-s + 1.79·21-s + (−3 − 5.19i)23-s + 9.37·25-s − 5·27-s + (1.10 + 1.91i)29-s − 31-s + ⋯
L(s)  = 1  + (−0.517 − 0.895i)3-s − 1.69·5-s + (−0.188 + 0.327i)7-s + (−0.0347 + 0.0602i)9-s + (0.571 + 0.989i)11-s + (0.693 + 0.720i)13-s + (0.876 + 1.51i)15-s + (−0.363 + 0.630i)17-s + (−0.845 + 1.46i)19-s + 0.390·21-s + (−0.625 − 1.08i)23-s + 1.87·25-s − 0.962·27-s + (0.205 + 0.355i)29-s − 0.179·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(364\)    =    \(2^{2} \cdot 7 \cdot 13\)
Sign: $0.0128 - 0.999i$
Analytic conductor: \(2.90655\)
Root analytic conductor: \(1.70486\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{364} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 364,\ (\ :1/2),\ 0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.304122 + 0.300246i\)
\(L(\frac12)\) \(\approx\) \(0.304122 + 0.300246i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-2.5 - 2.59i)T \)
good3 \( 1 + (0.895 + 1.55i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 3.79T + 5T^{2} \)
11 \( 1 + (-1.89 - 3.28i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.68 - 6.38i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.10 - 1.91i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.791 - 1.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.18 - 3.78i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 + 10.5T + 53T^{2} \)
59 \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.79 - 6.56i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 + (-2.68 - 4.65i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.18 + 15.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.75478887042410449066470024488, −11.14519595795780124538184014377, −9.875970960468925520630866234117, −8.516076255122149646769555655149, −7.921293429880098905389763300372, −6.74339628268830533111895161162, −6.32354088944258437282584434203, −4.47224557371159307375186829560, −3.72126341657009647447545791621, −1.65732336397060433724793382099, 0.31962908619490817082430572195, 3.33785266248429629243502448031, 4.08097707525387481797908406776, 5.02040077863399731901928291529, 6.36788658552930167537781036919, 7.52071530922316272284849211141, 8.389949470308854772880028733866, 9.357618311228356701398399246132, 10.62150186122591898182108291972, 11.25690332274593772350584002423

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