Properties

Label 2-1110-37.36-c1-0-20
Degree $2$
Conductor $1110$
Sign $-0.944 + 0.328i$
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  i·2-s − 3-s − 4-s i·5-s + i·6-s − 2.37·7-s + i·8-s + 9-s − 10-s + 6.37·11-s + 12-s − 4.74i·13-s + 2.37i·14-s + i·15-s + 16-s + 0.372i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.447i·5-s + 0.408i·6-s − 0.896·7-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s + 1.92·11-s + 0.288·12-s − 1.31i·13-s + 0.634i·14-s + 0.258i·15-s + 0.250·16-s + 0.0902i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.944 + 0.328i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.944 + 0.328i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8756459534\)
\(L(\frac12)\) \(\approx\) \(0.8756459534\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + T \)
5 \( 1 + iT \)
37 \( 1 + (5.74 - 2i)T \)
good7 \( 1 + 2.37T + 7T^{2} \)
11 \( 1 - 6.37T + 11T^{2} \)
13 \( 1 + 4.74iT - 13T^{2} \)
17 \( 1 - 0.372iT - 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + 8.74iT - 23T^{2} \)
29 \( 1 - 3.62iT - 29T^{2} \)
31 \( 1 + 8.37iT - 31T^{2} \)
41 \( 1 + 9.11T + 41T^{2} \)
43 \( 1 + 4.37iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 8.37T + 53T^{2} \)
59 \( 1 + 8iT - 59T^{2} \)
61 \( 1 + 5.62iT - 61T^{2} \)
67 \( 1 - 8.74T + 67T^{2} \)
71 \( 1 + 8.74T + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 + 6.74iT - 79T^{2} \)
83 \( 1 + 5.48T + 83T^{2} \)
89 \( 1 - 2.74iT - 89T^{2} \)
97 \( 1 + 7.11iT - 97T^{2} \)
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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

−9.739147080595628897344710903551, −8.811266976931953206973989181063, −8.051740612086504716818172397811, −6.67405373318571034917760811550, −6.12885641232563517508730735594, −5.09386012170125404188088093337, −3.99619345371932733534873309141, −3.31830088946845046858515595685, −1.70082284522225041213890629152, −0.44960265245632305675554817738, 1.48072770507138631093399316699, 3.34450951815330833025177302883, 4.15325665849173679356118180478, 5.20677988812925195523040751338, 6.39476317163920483779077110161, 6.69653103181398152554142283324, 7.27616659400631557910487840022, 8.782168798445302696870582822292, 9.352407388670562988935617017156, 9.901112283237803449318196883444

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