Properties

Label 2-1110-185.68-c1-0-18
Degree $2$
Conductor $1110$
Sign $0.878 + 0.477i$
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 + (1.94 − 1.11i)5-s + (−0.707 + 0.707i)6-s + (0.636 − 0.636i)7-s − 8-s − 1.00i·9-s + (−1.94 + 1.11i)10-s + 2.26i·11-s + (0.707 − 0.707i)12-s + 1.29·13-s + (−0.636 + 0.636i)14-s + (0.586 − 2.15i)15-s + 16-s + 5.74i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.408 − 0.408i)3-s + 0.5·4-s + (0.867 − 0.496i)5-s + (−0.288 + 0.288i)6-s + (0.240 − 0.240i)7-s − 0.353·8-s − 0.333i·9-s + (−0.613 + 0.351i)10-s + 0.684i·11-s + (0.204 − 0.204i)12-s + 0.358·13-s + (−0.170 + 0.170i)14-s + (0.151 − 0.557i)15-s + 0.250·16-s + 1.39i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.878 + 0.477i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.878 + 0.477i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.730024452\)
\(L(\frac12)\) \(\approx\) \(1.730024452\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.94 + 1.11i)T \)
37 \( 1 + (-3.16 + 5.19i)T \)
good7 \( 1 + (-0.636 + 0.636i)T - 7iT^{2} \)
11 \( 1 - 2.26iT - 11T^{2} \)
13 \( 1 - 1.29T + 13T^{2} \)
17 \( 1 - 5.74iT - 17T^{2} \)
19 \( 1 + (-4.10 - 4.10i)T + 19iT^{2} \)
23 \( 1 - 8.53T + 23T^{2} \)
29 \( 1 + (-1.70 + 1.70i)T - 29iT^{2} \)
31 \( 1 + (5.81 + 5.81i)T + 31iT^{2} \)
41 \( 1 - 9.02iT - 41T^{2} \)
43 \( 1 - 0.853T + 43T^{2} \)
47 \( 1 + (0.326 - 0.326i)T - 47iT^{2} \)
53 \( 1 + (6.79 + 6.79i)T + 53iT^{2} \)
59 \( 1 + (3.51 + 3.51i)T + 59iT^{2} \)
61 \( 1 + (2.06 + 2.06i)T + 61iT^{2} \)
67 \( 1 + (3.98 + 3.98i)T + 67iT^{2} \)
71 \( 1 + 15.2T + 71T^{2} \)
73 \( 1 + (-11.8 + 11.8i)T - 73iT^{2} \)
79 \( 1 + (9.04 + 9.04i)T + 79iT^{2} \)
83 \( 1 + (-9.12 - 9.12i)T + 83iT^{2} \)
89 \( 1 + (-6.32 + 6.32i)T - 89iT^{2} \)
97 \( 1 + 9.28iT - 97T^{2} \)
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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

−9.520083660296484317954955031367, −9.127401475536622120301789879447, −8.085979673868236697358448844339, −7.57246205812799359498927459491, −6.46788321367555612894610909439, −5.77392776494951748817662223959, −4.59664405452226296588592678487, −3.28731561092173643481385461103, −1.96161532278291641835790662036, −1.21968258777845293832580362794, 1.20769676042619987643468196025, 2.70548990018859394493828399165, 3.19329280657750226196687490561, 4.95578992136108067867161527828, 5.60115427268893846038045838340, 6.86197705761297626061634497545, 7.32504304640258806280856997239, 8.639594057303288110236456656807, 9.111015632285411691905856535578, 9.680697177208822419189431755196

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