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Properties

Label 2-1110-111.80-c1-0-22
Degree $2$
Conductor $1110$
Sign $0.991 - 0.132i$
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 + (−1.64 + 0.535i)3-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (−1.54 − 0.786i)6-s − 2.53·7-s + (−0.707 + 0.707i)8-s + (2.42 − 1.76i)9-s + 1.00·10-s + 1.55·11-s + (−0.535 − 1.64i)12-s + (−1.11 − 1.11i)13-s + (−1.79 − 1.79i)14-s + (−0.786 + 1.54i)15-s − 1.00·16-s + (4.84 − 4.84i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.951 + 0.309i)3-s + 0.500i·4-s + (0.316 − 0.316i)5-s + (−0.630 − 0.320i)6-s − 0.959·7-s + (−0.250 + 0.250i)8-s + (0.808 − 0.587i)9-s + 0.316·10-s + 0.468·11-s + (−0.154 − 0.475i)12-s + (−0.309 − 0.309i)13-s + (−0.479 − 0.479i)14-s + (−0.203 + 0.398i)15-s − 0.250·16-s + (1.17 − 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.991 - 0.132i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.991 - 0.132i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.416741798\)
\(L(\frac12)\) \(\approx\) \(1.416741798\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 - 0.535i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + (-3.85 - 4.70i)T \)
good7 \( 1 + 2.53T + 7T^{2} \)
11 \( 1 - 1.55T + 11T^{2} \)
13 \( 1 + (1.11 + 1.11i)T + 13iT^{2} \)
17 \( 1 + (-4.84 + 4.84i)T - 17iT^{2} \)
19 \( 1 + (2.49 + 2.49i)T + 19iT^{2} \)
23 \( 1 + (-0.856 + 0.856i)T - 23iT^{2} \)
29 \( 1 + (-5.19 - 5.19i)T + 29iT^{2} \)
31 \( 1 + (-1.70 + 1.70i)T - 31iT^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 + (2.78 + 2.78i)T + 43iT^{2} \)
47 \( 1 + 10.4iT - 47T^{2} \)
53 \( 1 - 5.16iT - 53T^{2} \)
59 \( 1 + (-8.36 + 8.36i)T - 59iT^{2} \)
61 \( 1 + (-7.41 + 7.41i)T - 61iT^{2} \)
67 \( 1 - 1.98iT - 67T^{2} \)
71 \( 1 + 3.37iT - 71T^{2} \)
73 \( 1 + 0.480iT - 73T^{2} \)
79 \( 1 + (-6.52 - 6.52i)T + 79iT^{2} \)
83 \( 1 - 8.64iT - 83T^{2} \)
89 \( 1 + (-10.5 - 10.5i)T + 89iT^{2} \)
97 \( 1 + (5.01 + 5.01i)T + 97iT^{2} \)
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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

−9.748118779348001952389677801024, −9.299130542376811448168292227715, −8.093544785198046866406945759383, −6.88658463292287244570577062358, −6.55102233723095943430787249938, −5.49473704946127817020488217686, −4.94291861896125863152974876882, −3.88059514514752542809261245660, −2.80280738872078937797513711268, −0.73258611373909099748991979405, 1.12551815573681457848317586353, 2.42404047874867614269868497049, 3.68501842863656162548321608174, 4.54495260196080434772497787919, 5.93094709919418570338974398825, 6.06507596638513949825603239024, 7.01694394060951991959084895122, 8.050429812771370832155706381410, 9.420835880719642957037475365806, 10.06704550951431509668456931389

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