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Properties

Label 2-74-37.27-c5-0-7
Degree $2$
Conductor $74$
Sign $0.817 + 0.575i$
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  + (−3.46 − 2i)2-s + (7.51 + 13.0i)3-s + (7.99 + 13.8i)4-s + (−14.6 + 8.47i)5-s − 60.1i·6-s + (−106. − 184. i)7-s − 63.9i·8-s + (8.59 − 14.8i)9-s + 67.7·10-s + 550.·11-s + (−120. + 208. i)12-s + (−233. + 134. i)13-s + 854. i·14-s + (−220. − 127. i)15-s + (−128 + 221. i)16-s + (1.21e3 + 703. i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.481 + 0.834i)3-s + (0.249 + 0.433i)4-s + (−0.262 + 0.151i)5-s − 0.681i·6-s + (−0.823 − 1.42i)7-s − 0.353i·8-s + (0.0353 − 0.0612i)9-s + 0.214·10-s + 1.37·11-s + (−0.240 + 0.417i)12-s + (−0.382 + 0.221i)13-s + 1.16i·14-s + (−0.253 − 0.146i)15-s + (−0.125 + 0.216i)16-s + (1.02 + 0.590i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.817 + 0.575i$
Analytic conductor: \(11.8684\)
Root analytic conductor: \(3.44505\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :5/2),\ 0.817 + 0.575i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.35300 - 0.428542i\)
\(L(\frac12)\) \(\approx\) \(1.35300 - 0.428542i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.46 + 2i)T \)
37 \( 1 + (-8.15e3 + 1.68e3i)T \)
good3 \( 1 + (-7.51 - 13.0i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (14.6 - 8.47i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (106. + 184. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 - 550.T + 1.61e5T^{2} \)
13 \( 1 + (233. - 134. i)T + (1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (-1.21e3 - 703. i)T + (7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-2.14e3 + 1.23e3i)T + (1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + 1.16e3iT - 6.43e6T^{2} \)
29 \( 1 + 5.80e3iT - 2.05e7T^{2} \)
31 \( 1 + 3.42e3iT - 2.86e7T^{2} \)
41 \( 1 + (7.60e3 + 1.31e4i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 - 1.52e4iT - 1.47e8T^{2} \)
47 \( 1 - 4.83e3T + 2.29e8T^{2} \)
53 \( 1 + (-1.32e3 + 2.29e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (4.47e4 + 2.58e4i)T + (3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-2.56e4 + 1.47e4i)T + (4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.39e4 + 2.42e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-9.21e3 - 1.59e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 - 6.66e4T + 2.07e9T^{2} \)
79 \( 1 + (-9.64e3 + 5.56e3i)T + (1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (1.66e4 - 2.89e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (2.33e4 + 1.34e4i)T + (2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 2.42e4iT - 8.58e9T^{2} \)
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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

−13.59218898194232305622015965488, −12.17257229940206318144959131338, −11.01088286115420818515862325887, −9.676040198578479086827995685770, −9.539928008032358092398352524971, −7.73880274850746948462032248139, −6.64100700697739787484674858779, −4.11395096976411220850374985662, −3.37340133179173342164024014411, −0.854511510585024933440836097848, 1.34899034413370966160494271464, 3.01415182342099508016995691065, 5.53536071641751453967285126490, 6.80998292225117793556261420708, 7.912505188596056489457310692656, 9.020186862196075553545220623312, 9.874819678540298939897853252098, 11.91762155014478237532159622384, 12.32011583548705434547004851270, 13.84515692535260952073434267365

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