Properties

Label 2-1110-185.68-c1-0-24
Degree $2$
Conductor $1110$
Sign $-0.0494 + 0.998i$
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 + (0.0687 − 2.23i)5-s + (0.707 − 0.707i)6-s + (2.13 − 2.13i)7-s − 8-s − 1.00i·9-s + (−0.0687 + 2.23i)10-s − 2.07i·11-s + (−0.707 + 0.707i)12-s + 4.31·13-s + (−2.13 + 2.13i)14-s + (1.53 + 1.62i)15-s + 16-s − 2.62i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.408 + 0.408i)3-s + 0.5·4-s + (0.0307 − 0.999i)5-s + (0.288 − 0.288i)6-s + (0.807 − 0.807i)7-s − 0.353·8-s − 0.333i·9-s + (−0.0217 + 0.706i)10-s − 0.625i·11-s + (−0.204 + 0.204i)12-s + 1.19·13-s + (−0.570 + 0.570i)14-s + (0.395 + 0.420i)15-s + 0.250·16-s − 0.635i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0494 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0494 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.0494 + 0.998i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.0494 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.018894433\)
\(L(\frac12)\) \(\approx\) \(1.018894433\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0687 + 2.23i)T \)
37 \( 1 + (2.65 + 5.47i)T \)
good7 \( 1 + (-2.13 + 2.13i)T - 7iT^{2} \)
11 \( 1 + 2.07iT - 11T^{2} \)
13 \( 1 - 4.31T + 13T^{2} \)
17 \( 1 + 2.62iT - 17T^{2} \)
19 \( 1 + (-1.48 - 1.48i)T + 19iT^{2} \)
23 \( 1 + 3.49T + 23T^{2} \)
29 \( 1 + (-0.364 + 0.364i)T - 29iT^{2} \)
31 \( 1 + (-0.391 - 0.391i)T + 31iT^{2} \)
41 \( 1 - 3.80iT - 41T^{2} \)
43 \( 1 + 6.28T + 43T^{2} \)
47 \( 1 + (-3.37 + 3.37i)T - 47iT^{2} \)
53 \( 1 + (1.87 + 1.87i)T + 53iT^{2} \)
59 \( 1 + (0.917 + 0.917i)T + 59iT^{2} \)
61 \( 1 + (-5.87 - 5.87i)T + 61iT^{2} \)
67 \( 1 + (3.05 + 3.05i)T + 67iT^{2} \)
71 \( 1 + 9.16T + 71T^{2} \)
73 \( 1 + (-8.50 + 8.50i)T - 73iT^{2} \)
79 \( 1 + (-9.01 - 9.01i)T + 79iT^{2} \)
83 \( 1 + (0.381 + 0.381i)T + 83iT^{2} \)
89 \( 1 + (-3.68 + 3.68i)T - 89iT^{2} \)
97 \( 1 + 14.1iT - 97T^{2} \)
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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

−9.636793202923704340704183147768, −8.730106582520800878460365017858, −8.211207137251172005931915411957, −7.35368080955553012750532897694, −6.17692703337205069887672594302, −5.39562113444800335419309105065, −4.41237199325060229734997977601, −3.50668248238437672190157792584, −1.63624663441532077123986649377, −0.64305608074203903170285694268, 1.52649281497860062234858793254, 2.41132342960635154780045274043, 3.71402068708995509828948929126, 5.14865774122871556608909564787, 6.11230780974780410853763743186, 6.69098594539301223867061152132, 7.69826649151391905465787360888, 8.292482419804086765742122185015, 9.174268848033929032602288663904, 10.24114187289576543913181127525

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