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Properties

Label 2-74-37.31-c4-0-9
Degree $2$
Conductor $74$
Sign $0.775 + 0.631i$
Analytic cond. $7.64937$
Root an. cond. $2.76575$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 2i)2-s − 9.86i·3-s + 8i·4-s + (0.419 − 0.419i)5-s + (19.7 − 19.7i)6-s + 41.6·7-s + (−16 + 16i)8-s − 16.2·9-s + 1.67·10-s − 228. i·11-s + 78.9·12-s + (148. − 148. i)13-s + (83.2 + 83.2i)14-s + (−4.13 − 4.13i)15-s − 64·16-s + (246. − 246. i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s − 1.09i·3-s + 0.5i·4-s + (0.0167 − 0.0167i)5-s + (0.547 − 0.547i)6-s + 0.849·7-s + (−0.250 + 0.250i)8-s − 0.201·9-s + 0.0167·10-s − 1.89i·11-s + 0.547·12-s + (0.881 − 0.881i)13-s + (0.424 + 0.424i)14-s + (−0.0183 − 0.0183i)15-s − 0.250·16-s + (0.852 − 0.852i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.775 + 0.631i$
Analytic conductor: \(7.64937\)
Root analytic conductor: \(2.76575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :2),\ 0.775 + 0.631i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.16550 - 0.769512i\)
\(L(\frac12)\) \(\approx\) \(2.16550 - 0.769512i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 - 2i)T \)
37 \( 1 + (987. + 948. i)T \)
good3 \( 1 + 9.86iT - 81T^{2} \)
5 \( 1 + (-0.419 + 0.419i)T - 625iT^{2} \)
7 \( 1 - 41.6T + 2.40e3T^{2} \)
11 \( 1 + 228. iT - 1.46e4T^{2} \)
13 \( 1 + (-148. + 148. i)T - 2.85e4iT^{2} \)
17 \( 1 + (-246. + 246. i)T - 8.35e4iT^{2} \)
19 \( 1 + (494. - 494. i)T - 1.30e5iT^{2} \)
23 \( 1 + (233. - 233. i)T - 2.79e5iT^{2} \)
29 \( 1 + (-914. - 914. i)T + 7.07e5iT^{2} \)
31 \( 1 + (-588. - 588. i)T + 9.23e5iT^{2} \)
41 \( 1 - 1.42e3iT - 2.82e6T^{2} \)
43 \( 1 + (-32.8 + 32.8i)T - 3.41e6iT^{2} \)
47 \( 1 + 2.78e3T + 4.87e6T^{2} \)
53 \( 1 - 2.54e3T + 7.89e6T^{2} \)
59 \( 1 + (1.65e3 - 1.65e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (76.4 + 76.4i)T + 1.38e7iT^{2} \)
67 \( 1 + 2.10e3iT - 2.01e7T^{2} \)
71 \( 1 - 8.19e3T + 2.54e7T^{2} \)
73 \( 1 - 1.70e3iT - 2.83e7T^{2} \)
79 \( 1 + (-526. + 526. i)T - 3.89e7iT^{2} \)
83 \( 1 + 7.30e3T + 4.74e7T^{2} \)
89 \( 1 + (-6.32e3 - 6.32e3i)T + 6.27e7iT^{2} \)
97 \( 1 + (4.62e3 - 4.62e3i)T - 8.85e7iT^{2} \)
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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

−13.76476027051385513143408943279, −12.83081145888457619628864766996, −11.78772464764565147537971409848, −10.66477698639238452511111961836, −8.430922096379086115298753837740, −7.929464214883911197684451410435, −6.43495869462532990073935428566, −5.43271966806804964946016692456, −3.38812507253055875028018114537, −1.20924063068972670625322754101, 1.99075078694443529883612269892, 4.22598631141783486991282888441, 4.67615674883907343523932518764, 6.57552643943204784590412727540, 8.440582340495826977924561353118, 9.828717499515770849739251531102, 10.52720128241146384711187080197, 11.67853613909705208618738083001, 12.74309053063889393361528698756, 14.11612065891895154083036197662

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