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Properties

Label 2-1110-185.117-c1-0-36
Degree $2$
Conductor $1110$
Sign $-0.511 + 0.859i$
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 + (1.62 − 1.53i)5-s + (−0.707 − 0.707i)6-s + (−2.68 − 2.68i)7-s + 8-s + 1.00i·9-s + (1.62 − 1.53i)10-s − 6.24i·11-s + (−0.707 − 0.707i)12-s + 4.65·13-s + (−2.68 − 2.68i)14-s + (−2.23 − 0.0651i)15-s + 16-s + 5.76i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.408 − 0.408i)3-s + 0.5·4-s + (0.727 − 0.686i)5-s + (−0.288 − 0.288i)6-s + (−1.01 − 1.01i)7-s + 0.353·8-s + 0.333i·9-s + (0.514 − 0.485i)10-s − 1.88i·11-s + (−0.204 − 0.204i)12-s + 1.29·13-s + (−0.717 − 0.717i)14-s + (−0.577 − 0.0168i)15-s + 0.250·16-s + 1.39i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.511 + 0.859i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.511 + 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.994874573\)
\(L(\frac12)\) \(\approx\) \(1.994874573\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 + 1.53i)T \)
37 \( 1 + (-1.16 + 5.96i)T \)
good7 \( 1 + (2.68 + 2.68i)T + 7iT^{2} \)
11 \( 1 + 6.24iT - 11T^{2} \)
13 \( 1 - 4.65T + 13T^{2} \)
17 \( 1 - 5.76iT - 17T^{2} \)
19 \( 1 + (4.77 - 4.77i)T - 19iT^{2} \)
23 \( 1 + 7.57T + 23T^{2} \)
29 \( 1 + (2.77 + 2.77i)T + 29iT^{2} \)
31 \( 1 + (-2.94 + 2.94i)T - 31iT^{2} \)
41 \( 1 - 2.10iT - 41T^{2} \)
43 \( 1 + 3.90T + 43T^{2} \)
47 \( 1 + (-3.89 - 3.89i)T + 47iT^{2} \)
53 \( 1 + (-7.12 + 7.12i)T - 53iT^{2} \)
59 \( 1 + (-2.06 + 2.06i)T - 59iT^{2} \)
61 \( 1 + (-8.65 + 8.65i)T - 61iT^{2} \)
67 \( 1 + (3.26 - 3.26i)T - 67iT^{2} \)
71 \( 1 - 5.79T + 71T^{2} \)
73 \( 1 + (-3.86 - 3.86i)T + 73iT^{2} \)
79 \( 1 + (-3.48 + 3.48i)T - 79iT^{2} \)
83 \( 1 + (-2.47 + 2.47i)T - 83iT^{2} \)
89 \( 1 + (0.594 + 0.594i)T + 89iT^{2} \)
97 \( 1 - 12.1iT - 97T^{2} \)
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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

−9.805412349557719079972331107240, −8.433070499079084564501546661340, −8.105728971213096704069369832389, −6.49847495766440799988868358920, −6.06461130149130092154898301490, −5.73608121022086412339120881355, −3.97332275322748099991984635713, −3.70601234879563357575450476052, −1.97135246165842932104871646996, −0.70763236922417939276702907960, 2.08045353881367424793918550227, 2.86165620273319426998576978281, 4.04093954109311167687960250967, 5.04819636165486124879006967686, 5.89185130709819027491364307446, 6.61229434107343854476635994848, 7.11981024752242258842389099250, 8.714988602345830221219565983943, 9.557430132841292936619005098753, 10.08084633212876885951564875140

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