Properties

Label 2-74-37.31-c4-0-13
Degree $2$
Conductor $74$
Sign $0.0665 - 0.997i$
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 − 16.1i·3-s + 8i·4-s + (−17.1 + 17.1i)5-s + (−32.2 + 32.2i)6-s − 18.7·7-s + (16 − 16i)8-s − 178.·9-s + 68.7·10-s + 90.7i·11-s + 128.·12-s + (49.3 − 49.3i)13-s + (37.4 + 37.4i)14-s + (276. + 276. i)15-s − 64·16-s + (208. − 208. i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s − 1.78i·3-s + 0.5i·4-s + (−0.687 + 0.687i)5-s + (−0.894 + 0.894i)6-s − 0.382·7-s + (0.250 − 0.250i)8-s − 2.20·9-s + 0.687·10-s + 0.749i·11-s + 0.894·12-s + (0.292 − 0.292i)13-s + (0.191 + 0.191i)14-s + (1.22 + 1.22i)15-s − 0.250·16-s + (0.721 − 0.721i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0665 - 0.997i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0665 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.0665 - 0.997i$
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: \((2,\ 74,\ (\ :2),\ 0.0665 - 0.997i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.000928968 + 0.000869070i\)
\(L(\frac12)\) \(\approx\) \(0.000928968 + 0.000869070i\)
\(L(3)\) 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 \( 1 + (-203. + 1.35e3i)T \)
good3 \( 1 + 16.1iT - 81T^{2} \)
5 \( 1 + (17.1 - 17.1i)T - 625iT^{2} \)
7 \( 1 + 18.7T + 2.40e3T^{2} \)
11 \( 1 - 90.7iT - 1.46e4T^{2} \)
13 \( 1 + (-49.3 + 49.3i)T - 2.85e4iT^{2} \)
17 \( 1 + (-208. + 208. i)T - 8.35e4iT^{2} \)
19 \( 1 + (166. - 166. i)T - 1.30e5iT^{2} \)
23 \( 1 + (686. - 686. i)T - 2.79e5iT^{2} \)
29 \( 1 + (-0.649 - 0.649i)T + 7.07e5iT^{2} \)
31 \( 1 + (1.08e3 + 1.08e3i)T + 9.23e5iT^{2} \)
41 \( 1 - 630. iT - 2.82e6T^{2} \)
43 \( 1 + (2.01e3 - 2.01e3i)T - 3.41e6iT^{2} \)
47 \( 1 - 2.15e3T + 4.87e6T^{2} \)
53 \( 1 + 3.42e3T + 7.89e6T^{2} \)
59 \( 1 + (2.15e3 - 2.15e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (4.32e3 + 4.32e3i)T + 1.38e7iT^{2} \)
67 \( 1 - 3.38e3iT - 2.01e7T^{2} \)
71 \( 1 + 2.13e3T + 2.54e7T^{2} \)
73 \( 1 + 9.33e3iT - 2.83e7T^{2} \)
79 \( 1 + (-1.61e3 + 1.61e3i)T - 3.89e7iT^{2} \)
83 \( 1 - 8.50e3T + 4.74e7T^{2} \)
89 \( 1 + (-6.12e3 - 6.12e3i)T + 6.27e7iT^{2} \)
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

−12.74512731600319687645090427162, −11.97232636242177481271062090301, −11.10428817771073282240900365855, −9.539200404837138174207629589068, −7.81668792611196115395363892034, −7.46873588631207887748914964112, −6.12121401380893787958098697361, −3.32084500459170972177967582874, −1.79460100112356694124601538047, −0.00075570786083443713575903326, 3.62138247176059017561337135656, 4.77036574554571497368864991280, 6.11816101841374599293925939697, 8.290929364209094442033459753246, 8.887394821655059196340150950431, 10.11206258819004973812412963434, 10.91576730372805932627389048763, 12.23734130957565561424566926493, 14.04035022068451785473119862044, 15.02059661986196961790416174588

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