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Properties

Label 2-1110-185.68-c1-0-29
Degree $2$
Conductor $1110$
Sign $0.322 + 0.946i$
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 + (−2.18 + 0.482i)5-s + (0.707 − 0.707i)6-s + (3.34 − 3.34i)7-s + 8-s − 1.00i·9-s + (−2.18 + 0.482i)10-s + 0.666i·11-s + (0.707 − 0.707i)12-s − 2.34·13-s + (3.34 − 3.34i)14-s + (−1.20 + 1.88i)15-s + 16-s − 2.53i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.408 − 0.408i)3-s + 0.5·4-s + (−0.976 + 0.215i)5-s + (0.288 − 0.288i)6-s + (1.26 − 1.26i)7-s + 0.353·8-s − 0.333i·9-s + (−0.690 + 0.152i)10-s + 0.200i·11-s + (0.204 − 0.204i)12-s − 0.651·13-s + (0.893 − 0.893i)14-s + (−0.310 + 0.486i)15-s + 0.250·16-s − 0.614i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.322 + 0.946i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.322 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.654837857\)
\(L(\frac12)\) \(\approx\) \(2.654837857\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.18 - 0.482i)T \)
37 \( 1 + (-5.08 + 3.34i)T \)
good7 \( 1 + (-3.34 + 3.34i)T - 7iT^{2} \)
11 \( 1 - 0.666iT - 11T^{2} \)
13 \( 1 + 2.34T + 13T^{2} \)
17 \( 1 + 2.53iT - 17T^{2} \)
19 \( 1 + (2.61 + 2.61i)T + 19iT^{2} \)
23 \( 1 - 7.61T + 23T^{2} \)
29 \( 1 + (-1.83 + 1.83i)T - 29iT^{2} \)
31 \( 1 + (3.00 + 3.00i)T + 31iT^{2} \)
41 \( 1 - 6.19iT - 41T^{2} \)
43 \( 1 + 9.47T + 43T^{2} \)
47 \( 1 + (5.52 - 5.52i)T - 47iT^{2} \)
53 \( 1 + (-3.00 - 3.00i)T + 53iT^{2} \)
59 \( 1 + (-1.77 - 1.77i)T + 59iT^{2} \)
61 \( 1 + (1.33 + 1.33i)T + 61iT^{2} \)
67 \( 1 + (-9.44 - 9.44i)T + 67iT^{2} \)
71 \( 1 - 3.44T + 71T^{2} \)
73 \( 1 + (-7.93 + 7.93i)T - 73iT^{2} \)
79 \( 1 + (8.95 + 8.95i)T + 79iT^{2} \)
83 \( 1 + (-6.18 - 6.18i)T + 83iT^{2} \)
89 \( 1 + (0.703 - 0.703i)T - 89iT^{2} \)
97 \( 1 - 6.06iT - 97T^{2} \)
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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

−9.769803117247982021636100501570, −8.557274406199446010290546404801, −7.73551282863523729236087706507, −7.28419873197127501170952348959, −6.61573475157871666573271693206, −4.92606186895919571735033456687, −4.54079168350584761200698423168, −3.52602203299756822733165044109, −2.42275419707143368824608929887, −0.943761104302597533599736501302, 1.76760870728189339453707125010, 2.90862953346006314295361477791, 3.90034533631142240211521683246, 4.96998735205100711102852846928, 5.24074196044849711098752442622, 6.63176153254869781022419841663, 7.67804969503993668461577309130, 8.453793217513690982319447322624, 8.816353546001462912608635001257, 10.11798184267678319207732363489

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