Properties

Label 2-74-1.1-c1-0-1
Degree $2$
Conductor $74$
Sign $1$
Analytic cond. $0.590892$
Root an. cond. $0.768695$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 1.61·3-s + 4-s + 3.85·5-s − 1.61·6-s − 3.23·7-s + 8-s − 0.381·9-s + 3.85·10-s − 1.38·11-s − 1.61·12-s − 2.85·13-s − 3.23·14-s − 6.23·15-s + 16-s − 4.47·17-s − 0.381·18-s + 4.47·19-s + 3.85·20-s + 5.23·21-s − 1.38·22-s + 2.85·23-s − 1.61·24-s + 9.85·25-s − 2.85·26-s + 5.47·27-s − 3.23·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.934·3-s + 0.5·4-s + 1.72·5-s − 0.660·6-s − 1.22·7-s + 0.353·8-s − 0.127·9-s + 1.21·10-s − 0.416·11-s − 0.467·12-s − 0.791·13-s − 0.864·14-s − 1.61·15-s + 0.250·16-s − 1.08·17-s − 0.0900·18-s + 1.02·19-s + 0.861·20-s + 1.14·21-s − 0.294·22-s + 0.595·23-s − 0.330·24-s + 1.97·25-s − 0.559·26-s + 1.05·27-s − 0.611·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $1$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.130442136\)
\(L(\frac12)\) \(\approx\) \(1.130442136\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
37 \( 1 + T \)
good3 \( 1 + 1.61T + 3T^{2} \)
5 \( 1 - 3.85T + 5T^{2} \)
7 \( 1 + 3.23T + 7T^{2} \)
11 \( 1 + 1.38T + 11T^{2} \)
13 \( 1 + 2.85T + 13T^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
19 \( 1 - 4.47T + 19T^{2} \)
23 \( 1 - 2.85T + 23T^{2} \)
29 \( 1 + 9.32T + 29T^{2} \)
31 \( 1 - 7.38T + 31T^{2} \)
41 \( 1 - 9.61T + 41T^{2} \)
43 \( 1 + 5.23T + 43T^{2} \)
47 \( 1 + 1.23T + 47T^{2} \)
53 \( 1 - 0.472T + 53T^{2} \)
59 \( 1 + 4.76T + 59T^{2} \)
61 \( 1 - 10.6T + 61T^{2} \)
67 \( 1 - 1.09T + 67T^{2} \)
71 \( 1 - 2.94T + 71T^{2} \)
73 \( 1 - 7.09T + 73T^{2} \)
79 \( 1 + 8.56T + 79T^{2} \)
83 \( 1 + 14.4T + 83T^{2} \)
89 \( 1 + 1.52T + 89T^{2} \)
97 \( 1 + 0.472T + 97T^{2} \)
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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

−14.31756017634910343877571717589, −13.30390780605588925601099910529, −12.72771702114412320601169767857, −11.36027416487763192359374783846, −10.18811168024303465092774942559, −9.334702209214140742115570595155, −6.87100223944678328516993531089, −5.97691423654313329530298681114, −5.12685721671521497213432134897, −2.68128995430614572206899555225, 2.68128995430614572206899555225, 5.12685721671521497213432134897, 5.97691423654313329530298681114, 6.87100223944678328516993531089, 9.334702209214140742115570595155, 10.18811168024303465092774942559, 11.36027416487763192359374783846, 12.72771702114412320601169767857, 13.30390780605588925601099910529, 14.31756017634910343877571717589

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