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Properties

Label 2-364-13.3-c1-0-4
Degree $2$
Conductor $364$
Sign $0.964 - 0.265i$
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  + (0.651 + 1.12i)3-s + 3.30·5-s + (0.5 − 0.866i)7-s + (0.651 − 1.12i)9-s + (−1.34 − 2.33i)11-s + (1.80 − 3.12i)13-s + (2.15 + 3.72i)15-s + (−2.80 + 4.85i)17-s + (−3.65 + 6.32i)19-s + 1.30·21-s + (−1 − 1.73i)23-s + 5.90·25-s + 5.60·27-s + (−1.65 − 2.86i)29-s − 7.60·31-s + ⋯
L(s)  = 1  + (0.376 + 0.651i)3-s + 1.47·5-s + (0.188 − 0.327i)7-s + (0.217 − 0.376i)9-s + (−0.406 − 0.704i)11-s + (0.499 − 0.866i)13-s + (0.555 + 0.962i)15-s + (−0.679 + 1.17i)17-s + (−0.837 + 1.45i)19-s + 0.284·21-s + (−0.208 − 0.361i)23-s + 1.18·25-s + 1.07·27-s + (−0.306 − 0.531i)29-s − 1.36·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(364\)    =    \(2^{2} \cdot 7 \cdot 13\)
Sign: $0.964 - 0.265i$
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: \((2,\ 364,\ (\ :1/2),\ 0.964 - 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87154 + 0.252500i\)
\(L(\frac12)\) \(\approx\) \(1.87154 + 0.252500i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.80 + 3.12i)T \)
good3 \( 1 + (-0.651 - 1.12i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3.30T + 5T^{2} \)
11 \( 1 + (1.34 + 2.33i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.80 - 4.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.65 - 6.32i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.65 + 2.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.60T + 31T^{2} \)
37 \( 1 + (-2.60 - 4.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.30 - 7.45i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.45 - 4.25i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.39T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 + (0.802 - 1.39i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.60 + 6.24i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.90 - 10.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.21T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
89 \( 1 + (7.95 + 13.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.45 + 14.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.05674326327765276513777601910, −10.36973136255884811855218918198, −9.802451245183555581019388913908, −8.773312102873286037005101642550, −7.998800979577138175609824982962, −6.26535634696070366813830564331, −5.85481695330638953558994276287, −4.38344948311692887120068685202, −3.24183421362289111243489981232, −1.70663876683441375589242825840, 1.87620254324778795540896787179, 2.45436803452976693604073378195, 4.58464912842067791282364016506, 5.56754909329035572350907897989, 6.77196669989586514359521555106, 7.38596674438009953330486377799, 8.944072970333037794852033398344, 9.237574453228264165104829861674, 10.47980634173440490488121171393, 11.28228913789705753615892666988

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