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Properties

Label 2-74-37.31-c6-0-15
Degree $2$
Conductor $74$
Sign $0.907 + 0.419i$
Analytic cond. $17.0240$
Root an. cond. $4.12601$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4 + 4i)2-s − 10.7i·3-s + 32i·4-s + (60.1 − 60.1i)5-s + (42.9 − 42.9i)6-s − 174.·7-s + (−128 + 128i)8-s + 613.·9-s + 480.·10-s − 1.12e3i·11-s + 343.·12-s + (739. − 739. i)13-s + (−696. − 696. i)14-s + (−645. − 645. i)15-s − 1.02e3·16-s + (1.31e3 − 1.31e3i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s − 0.397i·3-s + 0.5i·4-s + (0.480 − 0.480i)5-s + (0.198 − 0.198i)6-s − 0.507·7-s + (−0.250 + 0.250i)8-s + 0.841·9-s + 0.480·10-s − 0.847i·11-s + 0.198·12-s + (0.336 − 0.336i)13-s + (−0.253 − 0.253i)14-s + (−0.191 − 0.191i)15-s − 0.250·16-s + (0.267 − 0.267i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.907 + 0.419i$
Analytic conductor: \(17.0240\)
Root analytic conductor: \(4.12601\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3),\ 0.907 + 0.419i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.57500 - 0.566629i\)
\(L(\frac12)\) \(\approx\) \(2.57500 - 0.566629i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4 - 4i)T \)
37 \( 1 + (-4.40e4 - 2.49e4i)T \)
good3 \( 1 + 10.7iT - 729T^{2} \)
5 \( 1 + (-60.1 + 60.1i)T - 1.56e4iT^{2} \)
7 \( 1 + 174.T + 1.17e5T^{2} \)
11 \( 1 + 1.12e3iT - 1.77e6T^{2} \)
13 \( 1 + (-739. + 739. i)T - 4.82e6iT^{2} \)
17 \( 1 + (-1.31e3 + 1.31e3i)T - 2.41e7iT^{2} \)
19 \( 1 + (-4.16e3 + 4.16e3i)T - 4.70e7iT^{2} \)
23 \( 1 + (-1.48e4 + 1.48e4i)T - 1.48e8iT^{2} \)
29 \( 1 + (2.01e3 + 2.01e3i)T + 5.94e8iT^{2} \)
31 \( 1 + (-1.07e4 - 1.07e4i)T + 8.87e8iT^{2} \)
41 \( 1 - 7.58e3iT - 4.75e9T^{2} \)
43 \( 1 + (-3.32e4 + 3.32e4i)T - 6.32e9iT^{2} \)
47 \( 1 + 3.33e4T + 1.07e10T^{2} \)
53 \( 1 + 2.48e5T + 2.21e10T^{2} \)
59 \( 1 + (-1.45e4 + 1.45e4i)T - 4.21e10iT^{2} \)
61 \( 1 + (5.53e4 + 5.53e4i)T + 5.15e10iT^{2} \)
67 \( 1 - 8.65e4iT - 9.04e10T^{2} \)
71 \( 1 + 2.81e5T + 1.28e11T^{2} \)
73 \( 1 - 6.45e5iT - 1.51e11T^{2} \)
79 \( 1 + (1.94e5 - 1.94e5i)T - 2.43e11iT^{2} \)
83 \( 1 + 9.43e5T + 3.26e11T^{2} \)
89 \( 1 + (7.05e4 + 7.05e4i)T + 4.96e11iT^{2} \)
97 \( 1 + (5.53e4 - 5.53e4i)T - 8.32e11iT^{2} \)
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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

−13.15261225101092194909948861877, −12.72549406457651620115619622532, −11.24019332055059962128943204976, −9.738212663719136950933080918177, −8.547237829084766963703493124955, −7.17462418468119846389572956223, −6.09468569082329543933021417355, −4.79743593189110236477601235367, −3.06343533613795540370986663539, −0.975399552779322818693552720406, 1.57885254323010488374371621720, 3.24926085772660472137140890126, 4.58315681609495874928791989295, 6.08401730512847018548036934761, 7.34592508963761526306260650643, 9.460243039958337630199691326023, 10.02318153880884902365911539390, 11.15779097021456880708825013108, 12.47716453093068545005891446122, 13.32125529622942530097908854317

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