Properties

Label 2-1110-185.68-c1-0-36
Degree $2$
Conductor $1110$
Sign $-0.875 + 0.482i$
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 + (−2.22 − 0.248i)5-s + (0.707 − 0.707i)6-s + (−2.88 + 2.88i)7-s + 8-s − 1.00i·9-s + (−2.22 − 0.248i)10-s − 4.93i·11-s + (0.707 − 0.707i)12-s − 1.83·13-s + (−2.88 + 2.88i)14-s + (−1.74 + 1.39i)15-s + 16-s − 2.74i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.408 − 0.408i)3-s + 0.5·4-s + (−0.993 − 0.111i)5-s + (0.288 − 0.288i)6-s + (−1.09 + 1.09i)7-s + 0.353·8-s − 0.333i·9-s + (−0.702 − 0.0787i)10-s − 1.48i·11-s + (0.204 − 0.204i)12-s − 0.507·13-s + (−0.772 + 0.772i)14-s + (−0.451 + 0.360i)15-s + 0.250·16-s − 0.665i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 + 0.482i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.875 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.875 + 0.482i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.875 + 0.482i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8810926190\)
\(L(\frac12)\) \(\approx\) \(0.8810926190\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 + 0.248i)T \)
37 \( 1 + (-5.38 - 2.83i)T \)
good7 \( 1 + (2.88 - 2.88i)T - 7iT^{2} \)
11 \( 1 + 4.93iT - 11T^{2} \)
13 \( 1 + 1.83T + 13T^{2} \)
17 \( 1 + 2.74iT - 17T^{2} \)
19 \( 1 + (5.44 + 5.44i)T + 19iT^{2} \)
23 \( 1 + 8.34T + 23T^{2} \)
29 \( 1 + (-0.469 + 0.469i)T - 29iT^{2} \)
31 \( 1 + (-2.48 - 2.48i)T + 31iT^{2} \)
41 \( 1 + 4.25iT - 41T^{2} \)
43 \( 1 + 5.61T + 43T^{2} \)
47 \( 1 + (0.544 - 0.544i)T - 47iT^{2} \)
53 \( 1 + (7.02 + 7.02i)T + 53iT^{2} \)
59 \( 1 + (-7.44 - 7.44i)T + 59iT^{2} \)
61 \( 1 + (-1.23 - 1.23i)T + 61iT^{2} \)
67 \( 1 + (5.74 + 5.74i)T + 67iT^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 + (5.94 - 5.94i)T - 73iT^{2} \)
79 \( 1 + (4.75 + 4.75i)T + 79iT^{2} \)
83 \( 1 + (0.579 + 0.579i)T + 83iT^{2} \)
89 \( 1 + (-2.52 + 2.52i)T - 89iT^{2} \)
97 \( 1 - 17.4iT - 97T^{2} \)
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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

−9.293544752485790975798128840320, −8.564563830088887243353774092686, −7.925628603897651345699225159406, −6.73318279530020908768863072505, −6.25492404269705541418284279905, −5.20189057823703669337367427527, −4.04657782954543394431168465420, −3.10187739532776001729175861331, −2.46725363945999767268653470207, −0.26663165534856990021972355105, 2.06081292017727190801873054767, 3.36977613736369156273229120141, 4.18740765242430048609728062306, 4.43496177966205006383067057556, 6.07641280048425006310526884587, 6.85328507721615771915400620066, 7.66875231545292215931267532785, 8.253542824728251846378975170778, 9.838999716138258674201765809405, 10.03044568593519318093626720279

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