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Properties

Label 2-1110-111.80-c1-0-40
Degree $2$
Conductor $1110$
Sign $-0.257 + 0.966i$
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 + (0.125 + 1.72i)3-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (1.13 − 1.31i)6-s + 0.386·7-s + (0.707 − 0.707i)8-s + (−2.96 + 0.433i)9-s − 1.00·10-s + 1.11·11-s + (−1.72 + 0.125i)12-s + (−4.63 − 4.63i)13-s + (−0.273 − 0.273i)14-s + (1.31 + 1.13i)15-s − 1.00·16-s + (0.585 − 0.585i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.0724 + 0.997i)3-s + 0.500i·4-s + (0.316 − 0.316i)5-s + (0.462 − 0.534i)6-s + 0.146·7-s + (0.250 − 0.250i)8-s + (−0.989 + 0.144i)9-s − 0.316·10-s + 0.336·11-s + (−0.498 + 0.0362i)12-s + (−1.28 − 1.28i)13-s + (−0.0731 − 0.0731i)14-s + (0.338 + 0.292i)15-s − 0.250·16-s + (0.141 − 0.141i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.257 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.257 + 0.966i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.257 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7414571628\)
\(L(\frac12)\) \(\approx\) \(0.7414571628\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.125 - 1.72i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + (-3.10 - 5.23i)T \)
good7 \( 1 - 0.386T + 7T^{2} \)
11 \( 1 - 1.11T + 11T^{2} \)
13 \( 1 + (4.63 + 4.63i)T + 13iT^{2} \)
17 \( 1 + (-0.585 + 0.585i)T - 17iT^{2} \)
19 \( 1 + (4.45 + 4.45i)T + 19iT^{2} \)
23 \( 1 + (-3.21 + 3.21i)T - 23iT^{2} \)
29 \( 1 + (6.53 + 6.53i)T + 29iT^{2} \)
31 \( 1 + (2.95 - 2.95i)T - 31iT^{2} \)
41 \( 1 - 3.90T + 41T^{2} \)
43 \( 1 + (-2.30 - 2.30i)T + 43iT^{2} \)
47 \( 1 + 5.16iT - 47T^{2} \)
53 \( 1 - 10.1iT - 53T^{2} \)
59 \( 1 + (-7.38 + 7.38i)T - 59iT^{2} \)
61 \( 1 + (-0.777 + 0.777i)T - 61iT^{2} \)
67 \( 1 - 2.69iT - 67T^{2} \)
71 \( 1 + 7.49iT - 71T^{2} \)
73 \( 1 + 9.26iT - 73T^{2} \)
79 \( 1 + (6.39 + 6.39i)T + 79iT^{2} \)
83 \( 1 - 0.150iT - 83T^{2} \)
89 \( 1 + (9.82 + 9.82i)T + 89iT^{2} \)
97 \( 1 + (4.44 + 4.44i)T + 97iT^{2} \)
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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

−9.643241338225770445046325397450, −9.022232619136938183701366020495, −8.235947045169503644859369109866, −7.36176152954782042347795890091, −6.08675818011925407516272442487, −5.03767958101165898609236006996, −4.40131267945377521057353042393, −3.13204459798038238442097589269, −2.29261227673108254870466818583, −0.36735699792860090702398240412, 1.55802689811656644252222264635, 2.36083006593014786291008432807, 3.92447022690609403459115632482, 5.29883150519119138339358229920, 6.07977978143429450384953334619, 7.03122178508796902294037324672, 7.34041979690070981544992475745, 8.366538580518597757055781187455, 9.202060106858927931874036651465, 9.778845438888927440199202922216

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