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Properties

Label 2-74-37.10-c5-0-1
Degree $2$
Conductor $74$
Sign $0.991 - 0.131i$
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 + (−12.0 − 20.8i)3-s + (−7.99 − 13.8i)4-s + (51.5 + 89.2i)5-s − 96.0·6-s + (102. + 177. i)7-s − 63.9·8-s + (−166. + 289. i)9-s + 412.·10-s − 395.·11-s + (−192. + 332. i)12-s + (368. + 638. i)13-s + 821.·14-s + (1.23e3 − 2.14e3i)15-s + (−128 + 221. i)16-s + (−47.1 + 81.6i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.770 − 1.33i)3-s + (−0.249 − 0.433i)4-s + (0.922 + 1.59i)5-s − 1.08·6-s + (0.792 + 1.37i)7-s − 0.353·8-s + (−0.687 + 1.19i)9-s + 1.30·10-s − 0.985·11-s + (−0.385 + 0.667i)12-s + (0.604 + 1.04i)13-s + 1.12·14-s + (1.42 − 2.46i)15-s + (−0.125 + 0.216i)16-s + (−0.0395 + 0.0684i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.131i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.991 - 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.991 - 0.131i$
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: \((2,\ 74,\ (\ :5/2),\ 0.991 - 0.131i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.69708 + 0.111907i\)
\(L(\frac12)\) \(\approx\) \(1.69708 + 0.111907i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-8.32e3 - 128. i)T \)
good3 \( 1 + (12.0 + 20.8i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-51.5 - 89.2i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-102. - 177. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + 395.T + 1.61e5T^{2} \)
13 \( 1 + (-368. - 638. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (47.1 - 81.6i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (200. + 348. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 - 3.29e3T + 6.43e6T^{2} \)
29 \( 1 + 3.95e3T + 2.05e7T^{2} \)
31 \( 1 + 4.75e3T + 2.86e7T^{2} \)
41 \( 1 + (1.20e3 + 2.09e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 - 1.44e4T + 1.47e8T^{2} \)
47 \( 1 - 1.88e4T + 2.29e8T^{2} \)
53 \( 1 + (1.72e4 - 2.99e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.85e4 + 3.22e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-242. - 419. i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.12e4 + 3.67e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-9.42e3 - 1.63e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 - 1.80e4T + 2.07e9T^{2} \)
79 \( 1 + (-2.88e3 - 4.98e3i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (2.43e4 - 4.22e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (-6.32e4 + 1.09e5i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 9.31e4T + 8.58e9T^{2} \)
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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

−13.43897898869824622632952106984, −12.51116318315839009567371521215, −11.19077608287073660957846777255, −11.02045221999726775917469957852, −9.194411178078696923565035924884, −7.39691789170221040547977429642, −6.21587988860568695616781174137, −5.46211590803669529261628841139, −2.59599592391765286319251888582, −1.79544833215923564423371867559, 0.75428941618298079551499485131, 4.10872783498372419998316827880, 5.07093811508045894247119702426, 5.65371442441588097153791144139, 7.80331732036949424536713797441, 9.073690069925990086591482410808, 10.25253155961573576971474369342, 11.07440580294708568302916227006, 12.85587084835894073685763111390, 13.44354610679846655414552812565

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