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Properties

Label 2-666-37.10-c1-0-7
Degree $2$
Conductor $666$
Sign $0.757 + 0.652i$
Analytic cond. $5.31803$
Root an. cond. $2.30608$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (2.15 + 3.73i)7-s − 0.999·8-s − 0.999·10-s + 4.31·11-s + (−1 − 1.73i)13-s + 4.31·14-s + (−0.5 + 0.866i)16-s + (1.65 − 2.87i)17-s + (3.15 + 5.47i)19-s + (−0.499 + 0.866i)20-s + (2.15 − 3.73i)22-s − 4·23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + (0.815 + 1.41i)7-s − 0.353·8-s − 0.316·10-s + 1.30·11-s + (−0.277 − 0.480i)13-s + 1.15·14-s + (−0.125 + 0.216i)16-s + (0.402 − 0.696i)17-s + (0.724 + 1.25i)19-s + (−0.111 + 0.193i)20-s + (0.460 − 0.797i)22-s − 0.834·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.757 + 0.652i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.757 + 0.652i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82939 - 0.679756i\)
\(L(\frac12)\) \(\approx\) \(1.82939 - 0.679756i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
37 \( 1 + (5.97 - 1.14i)T \)
good5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.15 - 3.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 4.31T + 11T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.65 + 2.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.15 - 5.47i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 9.63T + 29T^{2} \)
31 \( 1 - 4.31T + 31T^{2} \)
41 \( 1 + (5.65 + 9.80i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 6.31T + 43T^{2} \)
47 \( 1 - 2.31T + 47T^{2} \)
53 \( 1 + (-4.31 + 7.47i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.31 - 4.01i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.65 - 6.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.15 + 5.47i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.15 - 5.47i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.633T + 73T^{2} \)
79 \( 1 + (-6.47 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.31 - 4.01i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.65 - 8.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.366T + 97T^{2} \)
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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

−10.39086625373749170597926985424, −9.663927268679427977829239357754, −8.612822581416929225313723013646, −8.206354343179522356591318724539, −6.70124185730592491019419282185, −5.59713490457440970944396125092, −4.95093767439934545922267358416, −3.80168941827729583107200529715, −2.56359744734095642705544406438, −1.31524018983352230055320536149, 1.29089172130969170223422834849, 3.28258758548054647280679636605, 4.25568231709419747125195768785, 4.93104153534609319858582970541, 6.48474844119719248625710136694, 6.94886536086931612280175369062, 7.81909425556487587515053674934, 8.667482613144015032057425885587, 9.780065545586167025845574864468, 10.64229655663125602165521158673

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