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Properties

Label 2-74-37.10-c5-0-10
Degree $2$
Conductor $74$
Sign $-0.548 + 0.835i$
Analytic cond. $11.8684$
Root an. cond. $3.44505$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 3.46i)2-s + (9.59 + 16.6i)3-s + (−7.99 − 13.8i)4-s + (−34.0 − 59.0i)5-s + 76.7·6-s + (−46.0 − 79.7i)7-s − 63.9·8-s + (−62.5 + 108. i)9-s − 272.·10-s − 135.·11-s + (153. − 265. i)12-s + (−248. − 431. i)13-s − 368.·14-s + (654. − 1.13e3i)15-s + (−128 + 221. i)16-s + (805. − 1.39e3i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.615 + 1.06i)3-s + (−0.249 − 0.433i)4-s + (−0.609 − 1.05i)5-s + 0.870·6-s + (−0.355 − 0.615i)7-s − 0.353·8-s + (−0.257 + 0.446i)9-s − 0.862·10-s − 0.336·11-s + (0.307 − 0.532i)12-s + (−0.408 − 0.707i)13-s − 0.502·14-s + (0.750 − 1.30i)15-s + (−0.125 + 0.216i)16-s + (0.675 − 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 + 0.835i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.548 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.548 + 0.835i$
Analytic conductor: \(11.8684\)
Root analytic conductor: \(3.44505\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :5/2),\ -0.548 + 0.835i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.769631 - 1.42596i\)
\(L(\frac12)\) \(\approx\) \(0.769631 - 1.42596i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 + 3.46i)T \)
37 \( 1 + (5.54e3 - 6.21e3i)T \)
good3 \( 1 + (-9.59 - 16.6i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (34.0 + 59.0i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (46.0 + 79.7i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + 135.T + 1.61e5T^{2} \)
13 \( 1 + (248. + 431. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (-805. + 1.39e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (282. + 488. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 - 641.T + 6.43e6T^{2} \)
29 \( 1 + 6.83e3T + 2.05e7T^{2} \)
31 \( 1 + 2.21e3T + 2.86e7T^{2} \)
41 \( 1 + (-4.96e3 - 8.60e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 - 1.69e4T + 1.47e8T^{2} \)
47 \( 1 - 1.61e4T + 2.29e8T^{2} \)
53 \( 1 + (1.17e4 - 2.02e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-3.88e3 + 6.72e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.92e4 + 3.32e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.34e4 - 2.32e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-249. - 432. i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 - 2.06e4T + 2.07e9T^{2} \)
79 \( 1 + (2.01e4 + 3.49e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-8.27e3 + 1.43e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (-7.11e3 + 1.23e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 1.05e5T + 8.58e9T^{2} \)
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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

−13.11900101552839115425792321159, −12.23174025108688876295845967166, −10.87211534731855623451459479298, −9.791802747098316288510394676740, −8.993065291280304138360082443418, −7.59940976317550997817761476210, −5.21582181641508535821017335500, −4.22849560162109678251463881151, −3.08789224019724965199822416949, −0.58231623272298007526018343242, 2.27264576096971052759011417288, 3.67197564170074883269479574298, 5.86918381587773015202975241085, 7.11386238431579920655207474417, 7.73320812398074802694180472612, 9.021584238417609144599343448262, 10.76986824799029966412096552078, 12.21095988591104846900340911802, 12.90685418944470379499185683036, 14.18305424018655386407086416626

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