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Properties

Label 2-74-37.26-c5-0-8
Degree $2$
Conductor $74$
Sign $0.254 - 0.967i$
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 + (−0.578 + 1.00i)3-s + (−7.99 + 13.8i)4-s + (40.4 − 69.9i)5-s − 4.62·6-s + (−85.9 + 148. i)7-s − 63.9·8-s + (120. + 209. i)9-s + 323.·10-s + 646.·11-s + (−9.25 − 16.0i)12-s + (176. − 305. i)13-s − 687.·14-s + (46.7 + 80.9i)15-s + (−128 − 221. i)16-s + (890. + 1.54e3i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.0370 + 0.0642i)3-s + (−0.249 + 0.433i)4-s + (0.722 − 1.25i)5-s − 0.0524·6-s + (−0.663 + 1.14i)7-s − 0.353·8-s + (0.497 + 0.861i)9-s + 1.02·10-s + 1.61·11-s + (−0.0185 − 0.0321i)12-s + (0.289 − 0.501i)13-s − 0.937·14-s + (0.0536 + 0.0928i)15-s + (−0.125 − 0.216i)16-s + (0.747 + 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.254 - 0.967i$
Analytic conductor: \(11.8684\)
Root analytic conductor: \(3.44505\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :5/2),\ 0.254 - 0.967i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.82783 + 1.40972i\)
\(L(\frac12)\) \(\approx\) \(1.82783 + 1.40972i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-912. + 8.27e3i)T \)
good3 \( 1 + (0.578 - 1.00i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-40.4 + 69.9i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (85.9 - 148. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 - 646.T + 1.61e5T^{2} \)
13 \( 1 + (-176. + 305. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (-890. - 1.54e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (1.13e3 - 1.95e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + 1.20e3T + 6.43e6T^{2} \)
29 \( 1 - 7.13e3T + 2.05e7T^{2} \)
31 \( 1 + 2.66e3T + 2.86e7T^{2} \)
41 \( 1 + (-6.20e3 + 1.07e4i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + 1.78e4T + 1.47e8T^{2} \)
47 \( 1 - 1.75e4T + 2.29e8T^{2} \)
53 \( 1 + (6.66e3 + 1.15e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (3.84e3 + 6.66e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (7.26e3 - 1.25e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.12e4 - 1.94e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-1.41e4 + 2.44e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 - 8.46e3T + 2.07e9T^{2} \)
79 \( 1 + (-4.43e4 + 7.68e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-1.99e4 - 3.46e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (6.57e4 + 1.13e5i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.37e5T + 8.58e9T^{2} \)
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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

−13.80377754440024171904993420538, −12.56211041452766587007863368560, −12.33134444058756500276859209126, −10.18163609675807283302553875962, −9.037550039851910443248387343297, −8.230809243013807969587854331214, −6.24716733556459809286315361021, −5.52625622735344333657402147054, −4.00931739397261904325440434161, −1.66973454303088079317951216632, 1.05365768419092556450884584278, 3.02021671808629146329333257716, 4.18630673468802387100866828685, 6.58993927411392143117208748482, 6.74738821917950175570939990821, 9.372738235232281654146185700250, 10.00375893076963319316599072327, 11.14088800738496520258155715319, 12.17571126082248548037527931014, 13.65172836480053275863775406801

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