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Properties

Label 2-1110-185.117-c1-0-14
Degree $2$
Conductor $1110$
Sign $0.222 - 0.974i$
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  + 2-s + (0.707 + 0.707i)3-s + 4-s + (−0.314 + 2.21i)5-s + (0.707 + 0.707i)6-s + (−0.376 − 0.376i)7-s + 8-s + 1.00i·9-s + (−0.314 + 2.21i)10-s + 1.28i·11-s + (0.707 + 0.707i)12-s + 5.12·13-s + (−0.376 − 0.376i)14-s + (−1.78 + 1.34i)15-s + 16-s − 2.31i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.408 + 0.408i)3-s + 0.5·4-s + (−0.140 + 0.990i)5-s + (0.288 + 0.288i)6-s + (−0.142 − 0.142i)7-s + 0.353·8-s + 0.333i·9-s + (−0.0994 + 0.700i)10-s + 0.387i·11-s + (0.204 + 0.204i)12-s + 1.42·13-s + (−0.100 − 0.100i)14-s + (−0.461 + 0.346i)15-s + 0.250·16-s − 0.560i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.222 - 0.974i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.222 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.823482423\)
\(L(\frac12)\) \(\approx\) \(2.823482423\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (0.314 - 2.21i)T \)
37 \( 1 + (3.20 - 5.17i)T \)
good7 \( 1 + (0.376 + 0.376i)T + 7iT^{2} \)
11 \( 1 - 1.28iT - 11T^{2} \)
13 \( 1 - 5.12T + 13T^{2} \)
17 \( 1 + 2.31iT - 17T^{2} \)
19 \( 1 + (3.87 - 3.87i)T - 19iT^{2} \)
23 \( 1 - 6.29T + 23T^{2} \)
29 \( 1 + (-1.15 - 1.15i)T + 29iT^{2} \)
31 \( 1 + (7.11 - 7.11i)T - 31iT^{2} \)
41 \( 1 + 2.12iT - 41T^{2} \)
43 \( 1 - 2.56T + 43T^{2} \)
47 \( 1 + (-2.64 - 2.64i)T + 47iT^{2} \)
53 \( 1 + (-0.572 + 0.572i)T - 53iT^{2} \)
59 \( 1 + (-1.04 + 1.04i)T - 59iT^{2} \)
61 \( 1 + (-3.44 + 3.44i)T - 61iT^{2} \)
67 \( 1 + (5.32 - 5.32i)T - 67iT^{2} \)
71 \( 1 - 4.38T + 71T^{2} \)
73 \( 1 + (9.81 + 9.81i)T + 73iT^{2} \)
79 \( 1 + (-11.3 + 11.3i)T - 79iT^{2} \)
83 \( 1 + (-11.5 + 11.5i)T - 83iT^{2} \)
89 \( 1 + (8.09 + 8.09i)T + 89iT^{2} \)
97 \( 1 - 8.07iT - 97T^{2} \)
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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

−10.39955668427008296008364954913, −9.180529966130528036843341806305, −8.376686520069458495337660459318, −7.31191242165280812053875692386, −6.68160500105427481233452262394, −5.78030165106734468886522280003, −4.68962169171401141935177671351, −3.63321385826593071677315476515, −3.15158055338094368006286495008, −1.82268387350758462679171060438, 1.00724588505944396555560464681, 2.28074430173193180610338790854, 3.57299692370162342035603233388, 4.27985367441297187827890482035, 5.45194734649917502859252388749, 6.14245061718251870926274634797, 7.09048932287370843439042257585, 8.117677541927880444416408644346, 8.779617825422188343164215975555, 9.362954899857308949499994062360

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