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Properties

Label 2-74-37.6-c6-0-9
Degree $2$
Conductor $74$
Sign $0.426 + 0.904i$
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 − 23.7i·3-s − 32i·4-s + (114. + 114. i)5-s + (−95.1 − 95.1i)6-s + 375.·7-s + (−128 − 128i)8-s + 163.·9-s + 913.·10-s − 159. i·11-s − 761.·12-s + (2.82e3 + 2.82e3i)13-s + (1.50e3 − 1.50e3i)14-s + (2.71e3 − 2.71e3i)15-s − 1.02e3·16-s + (−2.32e3 − 2.32e3i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s − 0.881i·3-s − 0.5i·4-s + (0.913 + 0.913i)5-s + (−0.440 − 0.440i)6-s + 1.09·7-s + (−0.250 − 0.250i)8-s + 0.223·9-s + 0.913·10-s − 0.119i·11-s − 0.440·12-s + (1.28 + 1.28i)13-s + (0.546 − 0.546i)14-s + (0.804 − 0.804i)15-s − 0.250·16-s + (−0.472 − 0.472i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.426 + 0.904i$
Analytic conductor: \(17.0240\)
Root analytic conductor: \(4.12601\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3),\ 0.426 + 0.904i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.78736 - 1.76812i\)
\(L(\frac12)\) \(\approx\) \(2.78736 - 1.76812i\)
\(L(4)\) 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 \( 1 + (-2.52e4 - 4.38e4i)T \)
good3 \( 1 + 23.7iT - 729T^{2} \)
5 \( 1 + (-114. - 114. i)T + 1.56e4iT^{2} \)
7 \( 1 - 375.T + 1.17e5T^{2} \)
11 \( 1 + 159. iT - 1.77e6T^{2} \)
13 \( 1 + (-2.82e3 - 2.82e3i)T + 4.82e6iT^{2} \)
17 \( 1 + (2.32e3 + 2.32e3i)T + 2.41e7iT^{2} \)
19 \( 1 + (-205. - 205. i)T + 4.70e7iT^{2} \)
23 \( 1 + (1.41e4 + 1.41e4i)T + 1.48e8iT^{2} \)
29 \( 1 + (-5.40e3 + 5.40e3i)T - 5.94e8iT^{2} \)
31 \( 1 + (9.20e3 - 9.20e3i)T - 8.87e8iT^{2} \)
41 \( 1 + 7.13e3iT - 4.75e9T^{2} \)
43 \( 1 + (3.29e4 + 3.29e4i)T + 6.32e9iT^{2} \)
47 \( 1 + 4.97e4T + 1.07e10T^{2} \)
53 \( 1 - 1.49e5T + 2.21e10T^{2} \)
59 \( 1 + (-1.99e5 - 1.99e5i)T + 4.21e10iT^{2} \)
61 \( 1 + (1.21e5 - 1.21e5i)T - 5.15e10iT^{2} \)
67 \( 1 + 3.49e5iT - 9.04e10T^{2} \)
71 \( 1 + 4.78e5T + 1.28e11T^{2} \)
73 \( 1 - 7.11e5iT - 1.51e11T^{2} \)
79 \( 1 + (3.85e5 + 3.85e5i)T + 2.43e11iT^{2} \)
83 \( 1 - 4.58e5T + 3.26e11T^{2} \)
89 \( 1 + (4.11e5 - 4.11e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (-2.14e5 - 2.14e5i)T + 8.32e11iT^{2} \)
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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

−13.48019317103064620544819012806, −11.99839938232311035596300549030, −11.13844286797286408831047133206, −10.08790509439974564507838266427, −8.518468512467125510308860328956, −6.90763304983736053494557478884, −6.08448187688260636688186663583, −4.32077053330268887009351549161, −2.32103198070478924206741740524, −1.43351008086663334632076110251, 1.54191338570876468481560588825, 3.87215934666549295364685202301, 5.05344697062082427541012780097, 5.87971636982341155532657197261, 7.894353494598361574131959670448, 8.912326070714738678761964716241, 10.13697401675627474511282224412, 11.28423515166260814169996724545, 12.84100113577762538737904724751, 13.52446417139088283138923415776

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