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Properties

Label 2-1110-111.80-c1-0-18
Degree $2$
Conductor $1110$
Sign $-0.622 - 0.782i$
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  + (0.707 + 0.707i)2-s + (0.912 + 1.47i)3-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (−0.396 + 1.68i)6-s + 0.363·7-s + (−0.707 + 0.707i)8-s + (−1.33 + 2.68i)9-s + 1.00·10-s − 1.23·11-s + (−1.47 + 0.912i)12-s + (2.54 + 2.54i)13-s + (0.257 + 0.257i)14-s + (1.68 + 0.396i)15-s − 1.00·16-s + (−2.79 + 2.79i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.526 + 0.850i)3-s + 0.500i·4-s + (0.316 − 0.316i)5-s + (−0.161 + 0.688i)6-s + 0.137·7-s + (−0.250 + 0.250i)8-s + (−0.445 + 0.895i)9-s + 0.316·10-s − 0.372·11-s + (−0.425 + 0.263i)12-s + (0.706 + 0.706i)13-s + (0.0687 + 0.0687i)14-s + (0.435 + 0.102i)15-s − 0.250·16-s + (−0.677 + 0.677i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.622 - 0.782i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.622 - 0.782i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.500272233\)
\(L(\frac12)\) \(\approx\) \(2.500272233\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-0.912 - 1.47i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + (-5.07 - 3.35i)T \)
good7 \( 1 - 0.363T + 7T^{2} \)
11 \( 1 + 1.23T + 11T^{2} \)
13 \( 1 + (-2.54 - 2.54i)T + 13iT^{2} \)
17 \( 1 + (2.79 - 2.79i)T - 17iT^{2} \)
19 \( 1 + (-1.40 - 1.40i)T + 19iT^{2} \)
23 \( 1 + (0.456 - 0.456i)T - 23iT^{2} \)
29 \( 1 + (-1.61 - 1.61i)T + 29iT^{2} \)
31 \( 1 + (-0.163 + 0.163i)T - 31iT^{2} \)
41 \( 1 - 0.545T + 41T^{2} \)
43 \( 1 + (-4.75 - 4.75i)T + 43iT^{2} \)
47 \( 1 - 1.60iT - 47T^{2} \)
53 \( 1 + 8.02iT - 53T^{2} \)
59 \( 1 + (-6.45 + 6.45i)T - 59iT^{2} \)
61 \( 1 + (-2.60 + 2.60i)T - 61iT^{2} \)
67 \( 1 - 4.94iT - 67T^{2} \)
71 \( 1 + 14.8iT - 71T^{2} \)
73 \( 1 - 1.60iT - 73T^{2} \)
79 \( 1 + (-2.32 - 2.32i)T + 79iT^{2} \)
83 \( 1 + 8.43iT - 83T^{2} \)
89 \( 1 + (-0.550 - 0.550i)T + 89iT^{2} \)
97 \( 1 + (0.870 + 0.870i)T + 97iT^{2} \)
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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

−9.979316737280021669268449862582, −9.230216441411396822954378623670, −8.441865228274002296824127113832, −7.88116277996500229520971830798, −6.62825672039183404139016110259, −5.79991222266710275394464249689, −4.85053904323452697900873048804, −4.16365368512673317150659748665, −3.18962076205442359338658448876, −1.93433609796128084533888756642, 0.894302081828367753424094118972, 2.28136544740319775205348897806, 2.95677731968640655899960813179, 4.06446826403464305577211108620, 5.34600684049795373347829257789, 6.14208917337733170487852465529, 7.00876568908532633339670792109, 7.85004035083025102317519750201, 8.769028534336756048585112999929, 9.532050758930222952994422089726

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