Properties

Label 2-1110-185.68-c1-0-0
Degree $2$
Conductor $1110$
Sign $-0.992 + 0.118i$
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 + (1.69 − 1.45i)5-s + (0.707 − 0.707i)6-s + (−0.764 + 0.764i)7-s − 8-s − 1.00i·9-s + (−1.69 + 1.45i)10-s + 4.86i·11-s + (−0.707 + 0.707i)12-s − 4.63·13-s + (0.764 − 0.764i)14-s + (−0.170 + 2.22i)15-s + 16-s − 5.18i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.408 + 0.408i)3-s + 0.5·4-s + (0.759 − 0.650i)5-s + (0.288 − 0.288i)6-s + (−0.288 + 0.288i)7-s − 0.353·8-s − 0.333i·9-s + (−0.536 + 0.460i)10-s + 1.46i·11-s + (−0.204 + 0.204i)12-s − 1.28·13-s + (0.204 − 0.204i)14-s + (−0.0441 + 0.575i)15-s + 0.250·16-s − 1.25i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.992 + 0.118i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.992 + 0.118i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.05680311191\)
\(L(\frac12)\) \(\approx\) \(0.05680311191\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 + 1.45i)T \)
37 \( 1 + (-0.720 + 6.03i)T \)
good7 \( 1 + (0.764 - 0.764i)T - 7iT^{2} \)
11 \( 1 - 4.86iT - 11T^{2} \)
13 \( 1 + 4.63T + 13T^{2} \)
17 \( 1 + 5.18iT - 17T^{2} \)
19 \( 1 + (1.84 + 1.84i)T + 19iT^{2} \)
23 \( 1 + 4.72T + 23T^{2} \)
29 \( 1 + (6.46 - 6.46i)T - 29iT^{2} \)
31 \( 1 + (-1.27 - 1.27i)T + 31iT^{2} \)
41 \( 1 - 3.55iT - 41T^{2} \)
43 \( 1 - 7.49T + 43T^{2} \)
47 \( 1 + (8.00 - 8.00i)T - 47iT^{2} \)
53 \( 1 + (5.64 + 5.64i)T + 53iT^{2} \)
59 \( 1 + (0.790 + 0.790i)T + 59iT^{2} \)
61 \( 1 + (5.24 + 5.24i)T + 61iT^{2} \)
67 \( 1 + (0.777 + 0.777i)T + 67iT^{2} \)
71 \( 1 - 0.599T + 71T^{2} \)
73 \( 1 + (1.07 - 1.07i)T - 73iT^{2} \)
79 \( 1 + (9.56 + 9.56i)T + 79iT^{2} \)
83 \( 1 + (11.2 + 11.2i)T + 83iT^{2} \)
89 \( 1 + (1.70 - 1.70i)T - 89iT^{2} \)
97 \( 1 + 6.03iT - 97T^{2} \)
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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

−9.923350447961982246428684956392, −9.494876559816842733846282309981, −9.087887609357727053077478936841, −7.73313904521956760663964738067, −7.04971047140815543403229865828, −6.07839855801539089270490498191, −5.08200096974788007754280462153, −4.50057846815516703744100927708, −2.73967373662642623750452445946, −1.77625217485563005401484448171, 0.03066742547287904068607518462, 1.70900757759135815252425444279, 2.72993190003027403132719143370, 3.94786124113719077103813624346, 5.64565706944475976346933111425, 6.06749566370261704527139170636, 6.88409515027813347142291563801, 7.80092953670352415746342762568, 8.495217207846090724336755152825, 9.668031721673924086343365639597

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