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Properties

Label 2-1110-185.117-c1-0-9
Degree $2$
Conductor $1110$
Sign $0.421 - 0.906i$
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 + (1.26 + 1.84i)5-s + (−0.707 − 0.707i)6-s + (1.64 + 1.64i)7-s + 8-s + 1.00i·9-s + (1.26 + 1.84i)10-s + 4.17i·11-s + (−0.707 − 0.707i)12-s − 1.20·13-s + (1.64 + 1.64i)14-s + (0.410 − 2.19i)15-s + 16-s − 0.221i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.408 − 0.408i)3-s + 0.5·4-s + (0.565 + 0.824i)5-s + (−0.288 − 0.288i)6-s + (0.620 + 0.620i)7-s + 0.353·8-s + 0.333i·9-s + (0.399 + 0.583i)10-s + 1.25i·11-s + (−0.204 − 0.204i)12-s − 0.332·13-s + (0.438 + 0.438i)14-s + (0.105 − 0.567i)15-s + 0.250·16-s − 0.0538i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.421 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.421 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.421 - 0.906i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.421 - 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.363512418\)
\(L(\frac12)\) \(\approx\) \(2.363512418\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.26 - 1.84i)T \)
37 \( 1 + (-6.07 + 0.388i)T \)
good7 \( 1 + (-1.64 - 1.64i)T + 7iT^{2} \)
11 \( 1 - 4.17iT - 11T^{2} \)
13 \( 1 + 1.20T + 13T^{2} \)
17 \( 1 + 0.221iT - 17T^{2} \)
19 \( 1 + (3.73 - 3.73i)T - 19iT^{2} \)
23 \( 1 + 4.24T + 23T^{2} \)
29 \( 1 + (5.45 + 5.45i)T + 29iT^{2} \)
31 \( 1 + (-1.77 + 1.77i)T - 31iT^{2} \)
41 \( 1 - 2.58iT - 41T^{2} \)
43 \( 1 - 7.69T + 43T^{2} \)
47 \( 1 + (-5.62 - 5.62i)T + 47iT^{2} \)
53 \( 1 + (4.45 - 4.45i)T - 53iT^{2} \)
59 \( 1 + (-3.90 + 3.90i)T - 59iT^{2} \)
61 \( 1 + (-0.901 + 0.901i)T - 61iT^{2} \)
67 \( 1 + (-10.3 + 10.3i)T - 67iT^{2} \)
71 \( 1 - 9.11T + 71T^{2} \)
73 \( 1 + (3.76 + 3.76i)T + 73iT^{2} \)
79 \( 1 + (-0.571 + 0.571i)T - 79iT^{2} \)
83 \( 1 + (-7.69 + 7.69i)T - 83iT^{2} \)
89 \( 1 + (-4.28 - 4.28i)T + 89iT^{2} \)
97 \( 1 + 8.59iT - 97T^{2} \)
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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

−10.09370214759596790113815565747, −9.415373631313290273282864156964, −7.940101965653733312752409392104, −7.46096375653556458169178928356, −6.33873106855403444828089954409, −5.91086476406171461506154953592, −4.89676866868517952047721613964, −3.96316056742192073913593801262, −2.36867865202934892374888025869, −1.95245765696620696025827639192, 0.865597469482095802853823233146, 2.31422725712745032153921124511, 3.77567055769463928190278075253, 4.52518735320563838626583608917, 5.36962723885621853617446266875, 5.99297247344009912558199741633, 6.99747022674516812492329855839, 8.120972552290340841055190039522, 8.860033571786580308381743597005, 9.794677058917044024630981501436

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