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Properties

Label 2-1110-111.80-c1-0-37
Degree $2$
Conductor $1110$
Sign $0.638 + 0.769i$
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 + (1.65 − 0.522i)3-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (−1.53 − 0.797i)6-s + 3.44·7-s + (0.707 − 0.707i)8-s + (2.45 − 1.72i)9-s − 1.00·10-s + 3.04·11-s + (0.522 + 1.65i)12-s + (0.571 + 0.571i)13-s + (−2.43 − 2.43i)14-s + (0.797 − 1.53i)15-s − 1.00·16-s + (−1.58 + 1.58i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.953 − 0.301i)3-s + 0.500i·4-s + (0.316 − 0.316i)5-s + (−0.627 − 0.325i)6-s + 1.30·7-s + (0.250 − 0.250i)8-s + (0.817 − 0.575i)9-s − 0.316·10-s + 0.918·11-s + (0.150 + 0.476i)12-s + (0.158 + 0.158i)13-s + (−0.650 − 0.650i)14-s + (0.205 − 0.396i)15-s − 0.250·16-s + (−0.383 + 0.383i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.638 + 0.769i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.638 + 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.282322339\)
\(L(\frac12)\) \(\approx\) \(2.282322339\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.65 + 0.522i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + (1.89 - 5.77i)T \)
good7 \( 1 - 3.44T + 7T^{2} \)
11 \( 1 - 3.04T + 11T^{2} \)
13 \( 1 + (-0.571 - 0.571i)T + 13iT^{2} \)
17 \( 1 + (1.58 - 1.58i)T - 17iT^{2} \)
19 \( 1 + (-2.90 - 2.90i)T + 19iT^{2} \)
23 \( 1 + (2.05 - 2.05i)T - 23iT^{2} \)
29 \( 1 + (1.83 + 1.83i)T + 29iT^{2} \)
31 \( 1 + (6.33 - 6.33i)T - 31iT^{2} \)
41 \( 1 + 8.00T + 41T^{2} \)
43 \( 1 + (-3.17 - 3.17i)T + 43iT^{2} \)
47 \( 1 + 7.90iT - 47T^{2} \)
53 \( 1 + 12.4iT - 53T^{2} \)
59 \( 1 + (-2.00 + 2.00i)T - 59iT^{2} \)
61 \( 1 + (-2.00 + 2.00i)T - 61iT^{2} \)
67 \( 1 - 9.46iT - 67T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 + 4.40iT - 73T^{2} \)
79 \( 1 + (8.20 + 8.20i)T + 79iT^{2} \)
83 \( 1 + 10.8iT - 83T^{2} \)
89 \( 1 + (9.90 + 9.90i)T + 89iT^{2} \)
97 \( 1 + (-2.40 - 2.40i)T + 97iT^{2} \)
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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

−9.668038675713324122436577690499, −8.691873903038055624555273312051, −8.435566349378507733904960606662, −7.50048935837679212150223308851, −6.66926546781642008664055198634, −5.32569925400125268562418937891, −4.19922212084774312913245507925, −3.37717216868738766810183843596, −1.84869930087288002593169734899, −1.46863346138774645322742768959, 1.44525349787106296404903379581, 2.43773270003166365813242170099, 3.84751002297120564604836889013, 4.76774175024441823227535994285, 5.72588990361678750476072279645, 6.96076053610725381906646311103, 7.55581226781732606954512548143, 8.365063717286047798884606658779, 9.146965360021321317805620568983, 9.572863355284279643128683522304

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