Properties

Label 2-1110-185.142-c1-0-31
Degree $2$
Conductor $1110$
Sign $0.973 + 0.227i$
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  + i·2-s + (0.707 + 0.707i)3-s − 4-s + (1.73 + 1.40i)5-s + (−0.707 + 0.707i)6-s + (−3.05 − 3.05i)7-s i·8-s + 1.00i·9-s + (−1.40 + 1.73i)10-s − 3.00i·11-s + (−0.707 − 0.707i)12-s − 6.48i·13-s + (3.05 − 3.05i)14-s + (0.230 + 2.22i)15-s + 16-s + 6.00·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (0.776 + 0.630i)5-s + (−0.288 + 0.288i)6-s + (−1.15 − 1.15i)7-s − 0.353i·8-s + 0.333i·9-s + (−0.445 + 0.548i)10-s − 0.906i·11-s + (−0.204 − 0.204i)12-s − 1.79i·13-s + (0.815 − 0.815i)14-s + (0.0594 + 0.574i)15-s + 0.250·16-s + 1.45·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.227i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.973 + 0.227i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.973 + 0.227i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.544883454\)
\(L(\frac12)\) \(\approx\) \(1.544883454\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-1.73 - 1.40i)T \)
37 \( 1 + (1.11 - 5.98i)T \)
good7 \( 1 + (3.05 + 3.05i)T + 7iT^{2} \)
11 \( 1 + 3.00iT - 11T^{2} \)
13 \( 1 + 6.48iT - 13T^{2} \)
17 \( 1 - 6.00T + 17T^{2} \)
19 \( 1 + (4.44 + 4.44i)T + 19iT^{2} \)
23 \( 1 + 4.23iT - 23T^{2} \)
29 \( 1 + (-1.51 + 1.51i)T - 29iT^{2} \)
31 \( 1 + (-1.07 - 1.07i)T + 31iT^{2} \)
41 \( 1 - 6.80iT - 41T^{2} \)
43 \( 1 + 11.7iT - 43T^{2} \)
47 \( 1 + (-5.31 - 5.31i)T + 47iT^{2} \)
53 \( 1 + (5.63 - 5.63i)T - 53iT^{2} \)
59 \( 1 + (4.46 + 4.46i)T + 59iT^{2} \)
61 \( 1 + (7.40 + 7.40i)T + 61iT^{2} \)
67 \( 1 + (2.42 - 2.42i)T - 67iT^{2} \)
71 \( 1 + 9.18T + 71T^{2} \)
73 \( 1 + (2.01 + 2.01i)T + 73iT^{2} \)
79 \( 1 + (-6.59 - 6.59i)T + 79iT^{2} \)
83 \( 1 + (0.481 - 0.481i)T - 83iT^{2} \)
89 \( 1 + (-4.35 + 4.35i)T - 89iT^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−9.969727118774664587670776934941, −8.992438310361471852325839722846, −8.075274474008902007904963480213, −7.30917178428405991909755071066, −6.36301147290367867264689506316, −5.83282631363217313565188611952, −4.67836721213126343255393734244, −3.32310207753556734007814357571, −3.00140115628791025539623894186, −0.65268690708845108270163781088, 1.62736463307161982091483534038, 2.25400028477110426241138483460, 3.44794326173010137851622069656, 4.53434575598787702632069528520, 5.73471910629282505068926974734, 6.31051223600918458713367304890, 7.44812575623697625802234551037, 8.623506926094815810032664173294, 9.264804851673810921722745060999, 9.634400556288620141691831509024

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