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Properties

Label 2-372-31.2-c1-0-5
Degree $2$
Conductor $372$
Sign $0.499 + 0.866i$
Analytic cond. $2.97043$
Root an. cond. $1.72349$
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  + (0.809 − 0.587i)3-s + 0.0622·5-s + (−0.862 − 2.65i)7-s + (0.309 − 0.951i)9-s + (0.106 + 0.326i)11-s + (3.84 − 2.79i)13-s + (0.0503 − 0.0365i)15-s + (1.85 − 5.71i)17-s + (0.449 + 0.326i)19-s + (−2.25 − 1.64i)21-s + (−2.07 + 6.38i)23-s − 4.99·25-s + (−0.309 − 0.951i)27-s + (7.02 + 5.10i)29-s + (3.66 − 4.19i)31-s + ⋯
L(s)  = 1  + (0.467 − 0.339i)3-s + 0.0278·5-s + (−0.326 − 1.00i)7-s + (0.103 − 0.317i)9-s + (0.0320 + 0.0984i)11-s + (1.06 − 0.774i)13-s + (0.0130 − 0.00944i)15-s + (0.450 − 1.38i)17-s + (0.103 + 0.0749i)19-s + (−0.492 − 0.358i)21-s + (−0.432 + 1.33i)23-s − 0.999·25-s + (−0.0594 − 0.183i)27-s + (1.30 + 0.947i)29-s + (0.658 − 0.752i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 372 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(372\)    =    \(2^{2} \cdot 3 \cdot 31\)
Sign: $0.499 + 0.866i$
Analytic conductor: \(2.97043\)
Root analytic conductor: \(1.72349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{372} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 372,\ (\ :1/2),\ 0.499 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33493 - 0.771280i\)
\(L(\frac12)\) \(\approx\) \(1.33493 - 0.771280i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (-3.66 + 4.19i)T \)
good5 \( 1 - 0.0622T + 5T^{2} \)
7 \( 1 + (0.862 + 2.65i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-0.106 - 0.326i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-3.84 + 2.79i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.85 + 5.71i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.449 - 0.326i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (2.07 - 6.38i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-7.02 - 5.10i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + 3.79T + 37T^{2} \)
41 \( 1 + (-4.09 - 2.97i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (5.15 + 3.74i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (6.49 - 4.71i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.54 - 7.83i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.800 - 0.581i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 - 1.52T + 67T^{2} \)
71 \( 1 + (2.45 - 7.55i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.0299 - 0.0921i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.52 - 13.9i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.00 + 2.91i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-2.65 - 8.15i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-1.92 - 5.93i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.26340944419717762014098337932, −10.17120736453452069984970103515, −9.528843637240633929304925097067, −8.283020725818584946893050462045, −7.53522766694041030566104517913, −6.62778117047542841132938630893, −5.43143911916367035276116284075, −3.95939403870806384603038863558, −3.00210448483545998724929444067, −1.11633582173372723113853231525, 1.99031961100506765360296178708, 3.35676573982784945528624683702, 4.45153540119455070602534370597, 5.88349385113899201703647573598, 6.56982889899951388220253333900, 8.326164569843087199383913848162, 8.527994649597981920958459280178, 9.723598157363354225841832168001, 10.46418763046627339571987496592, 11.61749705966516957608634901152

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