Properties

Label 2-74-1.1-c1-0-2
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 + 3.30·3-s + 4-s − 2.30·5-s − 3.30·6-s − 2.60·7-s − 8-s + 7.90·9-s + 2.30·10-s − 2.30·11-s + 3.30·12-s + 1.30·13-s + 2.60·14-s − 7.60·15-s + 16-s − 6·17-s − 7.90·18-s + 2·19-s − 2.30·20-s − 8.60·21-s + 2.30·22-s + 3.90·23-s − 3.30·24-s + 0.302·25-s − 1.30·26-s + 16.2·27-s − 2.60·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.90·3-s + 0.5·4-s − 1.02·5-s − 1.34·6-s − 0.984·7-s − 0.353·8-s + 2.63·9-s + 0.728·10-s − 0.694·11-s + 0.953·12-s + 0.361·13-s + 0.696·14-s − 1.96·15-s + 0.250·16-s − 1.45·17-s − 1.86·18-s + 0.458·19-s − 0.514·20-s − 1.87·21-s + 0.490·22-s + 0.814·23-s − 0.674·24-s + 0.0605·25-s − 0.255·26-s + 3.11·27-s − 0.492·28-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

\(L(1)\) \(\approx\) \(0.9642948923\)
\(L(\frac12)\) \(\approx\) \(0.9642948923\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.30T + 3T^{2} \)
5 \( 1 + 2.30T + 5T^{2} \)
7 \( 1 + 2.60T + 7T^{2} \)
11 \( 1 + 2.30T + 11T^{2} \)
13 \( 1 - 1.30T + 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 3.90T + 23T^{2} \)
29 \( 1 + 3.90T + 29T^{2} \)
31 \( 1 + 0.302T + 31T^{2} \)
41 \( 1 - 9.90T + 41T^{2} \)
43 \( 1 - 0.605T + 43T^{2} \)
47 \( 1 - 4.60T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 10.6T + 59T^{2} \)
61 \( 1 - 7.51T + 61T^{2} \)
67 \( 1 + 3.51T + 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 - 9.11T + 79T^{2} \)
83 \( 1 - 2.78T + 83T^{2} \)
89 \( 1 + 9.21T + 89T^{2} \)
97 \( 1 + 16.4T + 97T^{2} \)
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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

−14.85715955519318499835389416333, −13.45578658411617519237182903232, −12.72332663292485060593600802018, −11.00558376992052385392883780627, −9.654858831283765204535950149774, −8.841253477501806828214993356905, −7.86588361361950631323811500192, −6.96682723610503755952759087238, −3.93060845224348257138198386426, −2.69741759734566796341285138193, 2.69741759734566796341285138193, 3.93060845224348257138198386426, 6.96682723610503755952759087238, 7.86588361361950631323811500192, 8.841253477501806828214993356905, 9.654858831283765204535950149774, 11.00558376992052385392883780627, 12.72332663292485060593600802018, 13.45578658411617519237182903232, 14.85715955519318499835389416333

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