Properties

Label 2-74-1.1-c3-0-3
Degree $2$
Conductor $74$
Sign $1$
Analytic cond. $4.36614$
Root an. cond. $2.08953$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 6.12·3-s + 4·4-s + 18.7·5-s − 12.2·6-s − 21.9·7-s − 8·8-s + 10.5·9-s − 37.4·10-s + 57.7·11-s + 24.5·12-s − 18.6·13-s + 43.8·14-s + 114.·15-s + 16·16-s − 44.5·17-s − 21.1·18-s + 128.·19-s + 74.9·20-s − 134.·21-s − 115.·22-s − 112.·23-s − 49.0·24-s + 226.·25-s + 37.2·26-s − 100.·27-s − 87.6·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.17·3-s + 0.5·4-s + 1.67·5-s − 0.833·6-s − 1.18·7-s − 0.353·8-s + 0.390·9-s − 1.18·10-s + 1.58·11-s + 0.589·12-s − 0.397·13-s + 0.836·14-s + 1.97·15-s + 0.250·16-s − 0.635·17-s − 0.276·18-s + 1.55·19-s + 0.838·20-s − 1.39·21-s − 1.11·22-s − 1.01·23-s − 0.416·24-s + 1.81·25-s + 0.281·26-s − 0.718·27-s − 0.591·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $1$
Analytic conductor: \(4.36614\)
Root analytic conductor: \(2.08953\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.786850710\)
\(L(\frac12)\) \(\approx\) \(1.786850710\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2T \)
37 \( 1 + 37T \)
good3 \( 1 - 6.12T + 27T^{2} \)
5 \( 1 - 18.7T + 125T^{2} \)
7 \( 1 + 21.9T + 343T^{2} \)
11 \( 1 - 57.7T + 1.33e3T^{2} \)
13 \( 1 + 18.6T + 2.19e3T^{2} \)
17 \( 1 + 44.5T + 4.91e3T^{2} \)
19 \( 1 - 128.T + 6.85e3T^{2} \)
23 \( 1 + 112.T + 1.21e4T^{2} \)
29 \( 1 + 127.T + 2.43e4T^{2} \)
31 \( 1 - 38.8T + 2.97e4T^{2} \)
41 \( 1 + 438.T + 6.89e4T^{2} \)
43 \( 1 + 256.T + 7.95e4T^{2} \)
47 \( 1 - 275.T + 1.03e5T^{2} \)
53 \( 1 - 0.358T + 1.48e5T^{2} \)
59 \( 1 - 698.T + 2.05e5T^{2} \)
61 \( 1 + 494.T + 2.26e5T^{2} \)
67 \( 1 + 525.T + 3.00e5T^{2} \)
71 \( 1 + 1.01e3T + 3.57e5T^{2} \)
73 \( 1 + 14.4T + 3.89e5T^{2} \)
79 \( 1 - 980.T + 4.93e5T^{2} \)
83 \( 1 + 273.T + 5.71e5T^{2} \)
89 \( 1 - 559.T + 7.04e5T^{2} \)
97 \( 1 - 77.1T + 9.12e5T^{2} \)
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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

−13.95618726829362024059731865348, −13.39178489842652739710856462010, −11.90054603684560371513123506662, −9.991396882815858988684437517264, −9.502657276706303082172623972497, −8.806655653872362639529637943363, −7.02332745766936583040110129147, −5.96372402165584654416789887286, −3.27368114020858683240359520402, −1.86114520505421590067682439614, 1.86114520505421590067682439614, 3.27368114020858683240359520402, 5.96372402165584654416789887286, 7.02332745766936583040110129147, 8.806655653872362639529637943363, 9.502657276706303082172623972497, 9.991396882815858988684437517264, 11.90054603684560371513123506662, 13.39178489842652739710856462010, 13.95618726829362024059731865348

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