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Properties

Label 2-1110-37.10-c1-0-26
Degree $2$
Conductor $1110$
Sign $-0.227 - 0.973i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
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 + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s − 0.999·6-s + (−1 − 1.73i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s − 0.999·10-s − 3·11-s + (−0.499 + 0.866i)12-s + (0.5 + 0.866i)13-s − 1.99·14-s + (−0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−1.5 + 2.59i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s − 0.408·6-s + (−0.377 − 0.654i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s − 0.316·10-s − 0.904·11-s + (−0.144 + 0.249i)12-s + (0.138 + 0.240i)13-s − 0.534·14-s + (−0.129 + 0.223i)15-s + (−0.125 + 0.216i)16-s + (−0.363 + 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.227 - 0.973i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.227 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2611622152\)
\(L(\frac12)\) \(\approx\) \(0.2611622152\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-5.5 - 2.59i)T \)
good7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
41 \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 16T + 73T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3 - 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−9.397475956183968375452256020224, −8.413891767465397692938730610832, −7.59516305255420728791166363044, −6.66182893324257744814483415743, −5.77346642013925864030653835475, −4.79779302550387108388619607525, −3.94455919470724546601306988437, −2.78970358859114631512576989945, −1.53349777772669168051512519892, −0.10303702758514352755249055477, 2.49018037182304158806041699831, 3.44398361314567617832430678531, 4.49089706044109309590591201849, 5.47147827991944199742244757752, 6.02259697832898749144219131149, 7.06407636279771638030903345979, 7.83911775193637718008306784699, 8.780544473229872773176726811034, 9.528598399655838776482392907883, 10.43128903120370529331587079626

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