Properties

Label 2-74-37.9-c1-0-0
Degree $2$
Conductor $74$
Sign $0.0780 - 0.996i$
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 + (−0.238 + 1.35i)3-s + (−0.939 + 0.342i)4-s + (0.266 + 0.223i)5-s − 1.37·6-s + (−0.365 − 0.307i)7-s + (−0.5 − 0.866i)8-s + (1.04 + 0.378i)9-s + (−0.173 + 0.300i)10-s + (−1.29 − 2.23i)11-s + (−0.238 − 1.35i)12-s + (4.21 − 1.53i)13-s + (0.238 − 0.413i)14-s + (−0.365 + 0.307i)15-s + (0.766 − 0.642i)16-s + (−1.88 − 0.687i)17-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (−0.137 + 0.782i)3-s + (−0.469 + 0.171i)4-s + (0.118 + 0.0998i)5-s − 0.561·6-s + (−0.138 − 0.116i)7-s + (−0.176 − 0.306i)8-s + (0.346 + 0.126i)9-s + (−0.0549 + 0.0951i)10-s + (−0.389 − 0.675i)11-s + (−0.0689 − 0.391i)12-s + (1.16 − 0.425i)13-s + (0.0638 − 0.110i)14-s + (−0.0944 + 0.0792i)15-s + (0.191 − 0.160i)16-s + (−0.458 − 0.166i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0780 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0780 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.0780 - 0.996i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ 0.0780 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.685204 + 0.633648i\)
\(L(\frac12)\) \(\approx\) \(0.685204 + 0.633648i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.42 - 5.57i)T \)
good3 \( 1 + (0.238 - 1.35i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (-0.266 - 0.223i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (0.365 + 0.307i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (1.29 + 2.23i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.21 + 1.53i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (1.88 + 0.687i)T + (13.0 + 10.9i)T^{2} \)
19 \( 1 + (-0.611 + 3.46i)T + (-17.8 - 6.49i)T^{2} \)
23 \( 1 + (2.60 - 4.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.114 + 0.197i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.74T + 31T^{2} \)
41 \( 1 + (10.0 - 3.66i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 - 8.85T + 43T^{2} \)
47 \( 1 + (3.72 - 6.45i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.35 + 2.81i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-3.43 + 2.87i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-1.35 + 0.491i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (8.06 + 6.76i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-1.54 + 8.74i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 - 7.18T + 73T^{2} \)
79 \( 1 + (-2.70 - 2.27i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (1.32 + 0.483i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-2.92 + 2.45i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (9.24 - 16.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−15.15957799162391824025842500137, −13.74554880250188375278344279024, −13.09319224226518596399961019961, −11.33535989188602221652010839108, −10.32423843176886395918193050457, −9.150952604979135287796908089041, −7.899241766303637650717508146373, −6.38630541976562303954617090626, −5.10691019732322619661000002747, −3.64938138511806075628993087682, 1.82942286564869793950187444219, 4.01169869595108353775807037165, 5.85147533168094710322640210041, 7.23027869145878622455091587293, 8.700973526744253656010853392608, 9.977628361005391274526845600193, 11.15337420931369318327016728051, 12.36269609609318604853904845256, 12.98146702662578851877470989330, 13.95736931101161466563019878702

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