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Properties

Label 2-1110-37.36-c1-0-12
Degree $2$
Conductor $1110$
Sign $0.986 + 0.164i$
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 − 3-s − 4-s + i·5-s − i·6-s − 7-s − i·8-s + 9-s − 10-s − 11-s + 12-s − i·13-s − i·14-s − i·15-s + 16-s − 5i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s − 0.377·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s − 0.277i·13-s − 0.267i·14-s − 0.258i·15-s + 0.250·16-s − 1.21i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.986 + 0.164i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.986 + 0.164i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9259428630\)
\(L(\frac12)\) \(\approx\) \(0.9259428630\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 - iT \)
37 \( 1 + (-6 - i)T \)
good7 \( 1 + T + 7T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + 5iT - 17T^{2} \)
19 \( 1 + 3iT - 19T^{2} \)
23 \( 1 + 3iT - 23T^{2} \)
29 \( 1 - 10iT - 29T^{2} \)
31 \( 1 + 2iT - 31T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 12iT - 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 - 11T + 53T^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 + 14iT - 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 - 10iT - 79T^{2} \)
83 \( 1 + 3T + 83T^{2} \)
89 \( 1 - 9iT - 89T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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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

−9.757674592322395651360084876713, −9.023074740307522681850076441493, −8.020486131892934550098755552225, −6.99416720499669562100549753595, −6.72400727434630523911025153399, −5.51968528325553463368885747218, −4.96389623615838547188196016628, −3.73114384413749356116416006896, −2.56023766661685018531833096121, −0.52536425682019317544191150121, 1.11520569429418792601494087728, 2.38677319813039334606711147299, 3.78866922353619549736702948144, 4.46116264552764782255529051814, 5.66328319385818833104268411075, 6.19429555039558421062594892004, 7.52301015595131319283184415607, 8.303894586725900257113948758249, 9.262152701564750930883455918575, 10.02592590252124287993361174379

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