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Properties

Label 2-364-13.4-c1-0-0
Degree $2$
Conductor $364$
Sign $-0.968 - 0.247i$
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  + (−0.888 + 1.53i)3-s + 3.82i·5-s + (−0.866 + 0.5i)7-s + (−0.0783 − 0.135i)9-s + (−3.74 − 2.16i)11-s + (−0.930 − 3.48i)13-s + (−5.89 − 3.40i)15-s + (1.45 + 2.52i)17-s + (4.72 − 2.72i)19-s − 1.77i·21-s + (−0.307 + 0.531i)23-s − 9.65·25-s − 5.05·27-s + (−4.62 + 8.01i)29-s + 5.30i·31-s + ⋯
L(s)  = 1  + (−0.512 + 0.888i)3-s + 1.71i·5-s + (−0.327 + 0.188i)7-s + (−0.0261 − 0.0452i)9-s + (−1.12 − 0.652i)11-s + (−0.258 − 0.966i)13-s + (−1.52 − 0.878i)15-s + (0.353 + 0.612i)17-s + (1.08 − 0.626i)19-s − 0.387i·21-s + (−0.0640 + 0.110i)23-s − 1.93·25-s − 0.972·27-s + (−0.859 + 1.48i)29-s + 0.953i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(364\)    =    \(2^{2} \cdot 7 \cdot 13\)
Sign: $-0.968 - 0.247i$
Analytic conductor: \(2.90655\)
Root analytic conductor: \(1.70486\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{364} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 364,\ (\ :1/2),\ -0.968 - 0.247i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.100807 + 0.802975i\)
\(L(\frac12)\) \(\approx\) \(0.100807 + 0.802975i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (0.930 + 3.48i)T \)
good3 \( 1 + (0.888 - 1.53i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 3.82iT - 5T^{2} \)
11 \( 1 + (3.74 + 2.16i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.45 - 2.52i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.72 + 2.72i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.307 - 0.531i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.62 - 8.01i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.30iT - 31T^{2} \)
37 \( 1 + (0.974 + 0.562i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-10.4 - 6.01i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.641 - 1.11i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.68iT - 47T^{2} \)
53 \( 1 + 3.96T + 53T^{2} \)
59 \( 1 + (6.68 - 3.85i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.17 - 12.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.468 - 0.270i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.96 + 4.60i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.36iT - 73T^{2} \)
79 \( 1 - 0.331T + 79T^{2} \)
83 \( 1 - 16.6iT - 83T^{2} \)
89 \( 1 + (-2.55 - 1.47i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.36 + 4.82i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.40052290442522041131849763258, −10.67733549842854904604161840450, −10.41156167140864396637728205269, −9.443655186998453973670996462282, −7.914214564899512005451310169447, −7.13697622873416781205482161970, −5.86702885474462720782009423184, −5.19214722502258749934648609852, −3.50185041604711151370773670927, −2.78996180662873795073372542301, 0.57612086924706609804743883379, 1.98825808549093292114532682479, 4.13085900920106347679584761838, 5.18528876908070036755889214502, 6.03323074727835598177216618952, 7.42376276737744644668533207862, 7.897919884940108151936021964574, 9.394433986692652157610156202158, 9.712459435104048457494736237226, 11.38031282462232941162189149744

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