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Properties

Label 2-1110-111.68-c1-0-34
Degree $2$
Conductor $1110$
Sign $0.433 + 0.901i$
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 + (1.73 − 0.0652i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (1.17 − 1.27i)6-s − 5.10·7-s + (−0.707 − 0.707i)8-s + (2.99 − 0.225i)9-s + 1.00·10-s + 2.52·11-s + (−0.0652 − 1.73i)12-s + (4.82 − 4.82i)13-s + (−3.60 + 3.60i)14-s + (1.27 + 1.17i)15-s − 1.00·16-s + (2.16 + 2.16i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.999 − 0.0376i)3-s − 0.500i·4-s + (0.316 + 0.316i)5-s + (0.480 − 0.518i)6-s − 1.92·7-s + (−0.250 − 0.250i)8-s + (0.997 − 0.0753i)9-s + 0.316·10-s + 0.760·11-s + (−0.0188 − 0.499i)12-s + (1.33 − 1.33i)13-s + (−0.964 + 0.964i)14-s + (0.327 + 0.304i)15-s − 0.250·16-s + (0.524 + 0.524i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.433 + 0.901i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.433 + 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.892742775\)
\(L(\frac12)\) \(\approx\) \(2.892742775\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 0.0652i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (5.45 + 2.68i)T \)
good7 \( 1 + 5.10T + 7T^{2} \)
11 \( 1 - 2.52T + 11T^{2} \)
13 \( 1 + (-4.82 + 4.82i)T - 13iT^{2} \)
17 \( 1 + (-2.16 - 2.16i)T + 17iT^{2} \)
19 \( 1 + (-2.84 + 2.84i)T - 19iT^{2} \)
23 \( 1 + (-1.47 - 1.47i)T + 23iT^{2} \)
29 \( 1 + (-4.98 + 4.98i)T - 29iT^{2} \)
31 \( 1 + (2.75 + 2.75i)T + 31iT^{2} \)
41 \( 1 + 4.87T + 41T^{2} \)
43 \( 1 + (6.39 - 6.39i)T - 43iT^{2} \)
47 \( 1 - 0.0691iT - 47T^{2} \)
53 \( 1 + 4.54iT - 53T^{2} \)
59 \( 1 + (9.35 + 9.35i)T + 59iT^{2} \)
61 \( 1 + (-4.43 - 4.43i)T + 61iT^{2} \)
67 \( 1 - 14.1iT - 67T^{2} \)
71 \( 1 - 1.36iT - 71T^{2} \)
73 \( 1 - 12.8iT - 73T^{2} \)
79 \( 1 + (3.36 - 3.36i)T - 79iT^{2} \)
83 \( 1 - 9.73iT - 83T^{2} \)
89 \( 1 + (7.24 - 7.24i)T - 89iT^{2} \)
97 \( 1 + (4.90 - 4.90i)T - 97iT^{2} \)
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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

−9.881602868546037328323995117730, −9.076199022807007463451144096374, −8.220276524816162386654737964000, −6.96947324778388142168048794003, −6.36608845612000153477051710512, −5.51598153113144371384114167754, −3.85322091469103386158999833143, −3.38055911879861155315605882121, −2.70467660795147792410367849322, −1.11361674383734673049775363997, 1.57006406611084154833127875439, 3.28674796535188695043640210836, 3.48018572600197384706373407503, 4.66587096616085512036114593217, 6.00401227962160967236001842222, 6.67929359313439896890481123763, 7.21140160778492204114395607658, 8.656129862559547938530498541934, 8.988058993123363298857484957679, 9.701847745025435468059387535904

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