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Properties

Label 2-74-37.10-c3-0-4
Degree $2$
Conductor $74$
Sign $0.597 + 0.802i$
Analytic cond. $4.36614$
Root an. cond. $2.08953$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (−1.13 − 1.96i)3-s + (−1.99 − 3.46i)4-s + (2.10 + 3.64i)5-s + 4.52·6-s + (−10.8 − 18.8i)7-s + 7.99·8-s + (10.9 − 18.9i)9-s − 8.42·10-s + 40.2·11-s + (−4.52 + 7.84i)12-s + (−36.3 − 62.9i)13-s + 43.5·14-s + (4.76 − 8.25i)15-s + (−8 + 13.8i)16-s + (47.5 − 82.3i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.217 − 0.377i)3-s + (−0.249 − 0.433i)4-s + (0.188 + 0.326i)5-s + 0.308·6-s + (−0.588 − 1.01i)7-s + 0.353·8-s + (0.405 − 0.701i)9-s − 0.266·10-s + 1.10·11-s + (−0.108 + 0.188i)12-s + (−0.775 − 1.34i)13-s + 0.832·14-s + (0.0820 − 0.142i)15-s + (−0.125 + 0.216i)16-s + (0.678 − 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 + 0.802i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.597 + 0.802i$
Analytic conductor: \(4.36614\)
Root analytic conductor: \(2.08953\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3/2),\ 0.597 + 0.802i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.913083 - 0.458609i\)
\(L(\frac12)\) \(\approx\) \(0.913083 - 0.458609i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
37 \( 1 + (-106. - 198. i)T \)
good3 \( 1 + (1.13 + 1.96i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-2.10 - 3.64i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (10.8 + 18.8i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 - 40.2T + 1.33e3T^{2} \)
13 \( 1 + (36.3 + 62.9i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-47.5 + 82.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-65.0 - 112. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 88.1T + 1.21e4T^{2} \)
29 \( 1 + 237.T + 2.43e4T^{2} \)
31 \( 1 + 138.T + 2.97e4T^{2} \)
41 \( 1 + (-136. - 235. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + 70.8T + 7.95e4T^{2} \)
47 \( 1 - 420.T + 1.03e5T^{2} \)
53 \( 1 + (-308. + 535. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (15.0 - 26.1i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-63.2 - 109. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-460. - 796. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (131. + 228. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 649.T + 3.89e5T^{2} \)
79 \( 1 + (164. + 285. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (179. - 310. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-7.22 + 12.5i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.23e3T + 9.12e5T^{2} \)
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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

−14.13005125479262740025823494425, −12.89741306430996386482896678224, −11.81212968363738026087821515293, −10.12143247311413254341369928452, −9.619932009502028521031686969997, −7.70118168067428178526241934750, −6.93178443801853886266156250495, −5.73534957619028468817560595270, −3.68754271223862249323571001576, −0.804525395352835630687511448152, 1.99121214621547992790247548151, 4.02511660668479867841496286188, 5.57241185521436466269864120629, 7.27776969195025247624816658080, 9.156942399208980452513521789948, 9.433652244706809661019119390124, 10.95580668263349650864444582347, 11.98636668459358143562476756287, 12.84054753719658588351457510761, 14.14187621430954032689212943578

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