Properties

Label 2-74-37.33-c1-0-1
Degree $2$
Conductor $74$
Sign $0.973 - 0.230i$
Analytic cond. $0.590892$
Root an. cond. $0.768695$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 − 0.984i)2-s + (0.504 + 2.86i)3-s + (−0.939 − 0.342i)4-s + (0.266 − 0.223i)5-s + 2.90·6-s + (0.773 − 0.649i)7-s + (−0.5 + 0.866i)8-s + (−5.12 + 1.86i)9-s + (−0.173 − 0.300i)10-s + (2.73 − 4.73i)11-s + (0.504 − 2.86i)12-s + (−5.97 − 2.17i)13-s + (−0.504 − 0.874i)14-s + (0.773 + 0.649i)15-s + (0.766 + 0.642i)16-s + (−0.490 + 0.178i)17-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (0.291 + 1.65i)3-s + (−0.469 − 0.171i)4-s + (0.118 − 0.0998i)5-s + 1.18·6-s + (0.292 − 0.245i)7-s + (−0.176 + 0.306i)8-s + (−1.70 + 0.621i)9-s + (−0.0549 − 0.0951i)10-s + (0.823 − 1.42i)11-s + (0.145 − 0.826i)12-s + (−1.65 − 0.603i)13-s + (−0.134 − 0.233i)14-s + (0.199 + 0.167i)15-s + (0.191 + 0.160i)16-s + (−0.119 + 0.0433i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.973 - 0.230i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ 0.973 - 0.230i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01343 + 0.118256i\)
\(L(\frac12)\) \(\approx\) \(1.01343 + 0.118256i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
37 \( 1 + (5.88 + 1.53i)T \)
good3 \( 1 + (-0.504 - 2.86i)T + (-2.81 + 1.02i)T^{2} \)
5 \( 1 + (-0.266 + 0.223i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-0.773 + 0.649i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-2.73 + 4.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.97 + 2.17i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.490 - 0.178i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (-0.869 - 4.93i)T + (-17.8 + 6.49i)T^{2} \)
23 \( 1 + (0.721 + 1.24i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.16 + 5.48i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.30T + 31T^{2} \)
41 \( 1 + (-1.03 - 0.376i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + 6.27T + 43T^{2} \)
47 \( 1 + (-5.71 - 9.90i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.0706 - 0.0593i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-2.29 - 1.92i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (3.58 + 1.30i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.892 - 0.748i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.0543 - 0.308i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 + (-7.12 + 5.98i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (8.89 - 3.23i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (12.0 + 10.1i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-0.464 - 0.804i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.50524290523407492192357866859, −13.88408091760935021859654418875, −12.17004443038072295400497930740, −11.08979969309910324723610034393, −10.14640485824757525784815307111, −9.357933956818691750759675334508, −8.172007026784280048170172057084, −5.62298570593166065922861843854, −4.37859637584662036455254527585, −3.15439325335823151430107622770, 2.19179960255726958692821388548, 4.89447097243458177858122036153, 6.82882691075811576949023221098, 7.08390411798279786996538289576, 8.464910518162430431752206534220, 9.690104649648977455411997334394, 11.97861067913557097953870966237, 12.32421374721966689575954077475, 13.63895973436123503488484599952, 14.40724414514435799414318761010

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