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Properties

Label 2-74-37.31-c4-0-3
Degree $2$
Conductor $74$
Sign $-0.0880 - 0.996i$
Analytic cond. $7.64937$
Root an. cond. $2.76575$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 2i)2-s + 12.3i·3-s + 8i·4-s + (10.7 − 10.7i)5-s + (24.6 − 24.6i)6-s + 35.0·7-s + (16 − 16i)8-s − 70.4·9-s − 43.0·10-s + 142. i·11-s − 98.4·12-s + (−181. + 181. i)13-s + (−70.1 − 70.1i)14-s + (132. + 132. i)15-s − 64·16-s + (−88.0 + 88.0i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + 1.36i·3-s + 0.5i·4-s + (0.430 − 0.430i)5-s + (0.683 − 0.683i)6-s + 0.715·7-s + (0.250 − 0.250i)8-s − 0.870·9-s − 0.430·10-s + 1.17i·11-s − 0.683·12-s + (−1.07 + 1.07i)13-s + (−0.357 − 0.357i)14-s + (0.588 + 0.588i)15-s − 0.250·16-s + (−0.304 + 0.304i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0880 - 0.996i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0880 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.0880 - 0.996i$
Analytic conductor: \(7.64937\)
Root analytic conductor: \(2.76575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :2),\ -0.0880 - 0.996i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.831535 + 0.908280i\)
\(L(\frac12)\) \(\approx\) \(0.831535 + 0.908280i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 + 2i)T \)
37 \( 1 + (7.65 + 1.36e3i)T \)
good3 \( 1 - 12.3iT - 81T^{2} \)
5 \( 1 + (-10.7 + 10.7i)T - 625iT^{2} \)
7 \( 1 - 35.0T + 2.40e3T^{2} \)
11 \( 1 - 142. iT - 1.46e4T^{2} \)
13 \( 1 + (181. - 181. i)T - 2.85e4iT^{2} \)
17 \( 1 + (88.0 - 88.0i)T - 8.35e4iT^{2} \)
19 \( 1 + (-13.8 + 13.8i)T - 1.30e5iT^{2} \)
23 \( 1 + (-79.7 + 79.7i)T - 2.79e5iT^{2} \)
29 \( 1 + (-420. - 420. i)T + 7.07e5iT^{2} \)
31 \( 1 + (624. + 624. i)T + 9.23e5iT^{2} \)
41 \( 1 - 1.23e3iT - 2.82e6T^{2} \)
43 \( 1 + (-607. + 607. i)T - 3.41e6iT^{2} \)
47 \( 1 - 2.97e3T + 4.87e6T^{2} \)
53 \( 1 - 1.04e3T + 7.89e6T^{2} \)
59 \( 1 + (-1.89e3 + 1.89e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (-4.01e3 - 4.01e3i)T + 1.38e7iT^{2} \)
67 \( 1 - 2.33e3iT - 2.01e7T^{2} \)
71 \( 1 - 3.64e3T + 2.54e7T^{2} \)
73 \( 1 + 9.94e3iT - 2.83e7T^{2} \)
79 \( 1 + (-6.40e3 + 6.40e3i)T - 3.89e7iT^{2} \)
83 \( 1 + 7.15e3T + 4.74e7T^{2} \)
89 \( 1 + (-2.25e3 - 2.25e3i)T + 6.27e7iT^{2} \)
97 \( 1 + (-8.55e3 + 8.55e3i)T - 8.85e7iT^{2} \)
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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

−14.38767916124205121306494080633, −12.79810947069668887105006603298, −11.65146014319163759982367065427, −10.55281544743404011233038173518, −9.597581784163030753516922524639, −8.974415434692646879142326846420, −7.30482887869905854809171451464, −5.05497803725509012840740769734, −4.18808379457413530833787960209, −2.01903410027451328849633560125, 0.76578567989378638861092152207, 2.45777318112433199143028461860, 5.40502016018711401965756541817, 6.59453978076945189708970205520, 7.64718106813488991655038210042, 8.503506752713963402867187098384, 10.14341723917353086614511595025, 11.33074331448218221021051120149, 12.55405301488950536203136524040, 13.74129327489472865246681997862

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