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Properties

Label 2-1110-111.80-c1-0-32
Degree $2$
Conductor $1110$
Sign $0.434 - 0.900i$
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 + (1.72 − 0.161i)3-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (1.33 + 1.10i)6-s + 0.212·7-s + (−0.707 + 0.707i)8-s + (2.94 − 0.556i)9-s − 1.00·10-s + 4.71·11-s + (0.161 + 1.72i)12-s + (1.41 + 1.41i)13-s + (0.150 + 0.150i)14-s + (−1.10 + 1.33i)15-s − 1.00·16-s + (−2.20 + 2.20i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.995 − 0.0931i)3-s + 0.500i·4-s + (−0.316 + 0.316i)5-s + (0.544 + 0.451i)6-s + 0.0804·7-s + (−0.250 + 0.250i)8-s + (0.982 − 0.185i)9-s − 0.316·10-s + 1.42·11-s + (0.0465 + 0.497i)12-s + (0.391 + 0.391i)13-s + (0.0402 + 0.0402i)14-s + (−0.285 + 0.344i)15-s − 0.250·16-s + (−0.534 + 0.534i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.434 - 0.900i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.434 - 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.071150105\)
\(L(\frac12)\) \(\approx\) \(3.071150105\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.161i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (-5.76 + 1.94i)T \)
good7 \( 1 - 0.212T + 7T^{2} \)
11 \( 1 - 4.71T + 11T^{2} \)
13 \( 1 + (-1.41 - 1.41i)T + 13iT^{2} \)
17 \( 1 + (2.20 - 2.20i)T - 17iT^{2} \)
19 \( 1 + (1.02 + 1.02i)T + 19iT^{2} \)
23 \( 1 + (1.60 - 1.60i)T - 23iT^{2} \)
29 \( 1 + (3.47 + 3.47i)T + 29iT^{2} \)
31 \( 1 + (-0.648 + 0.648i)T - 31iT^{2} \)
41 \( 1 - 1.68T + 41T^{2} \)
43 \( 1 + (-2.65 - 2.65i)T + 43iT^{2} \)
47 \( 1 - 12.0iT - 47T^{2} \)
53 \( 1 - 6.21iT - 53T^{2} \)
59 \( 1 + (-5.08 + 5.08i)T - 59iT^{2} \)
61 \( 1 + (0.209 - 0.209i)T - 61iT^{2} \)
67 \( 1 + 6.62iT - 67T^{2} \)
71 \( 1 - 9.09iT - 71T^{2} \)
73 \( 1 + 7.72iT - 73T^{2} \)
79 \( 1 + (12.0 + 12.0i)T + 79iT^{2} \)
83 \( 1 - 1.76iT - 83T^{2} \)
89 \( 1 + (-6.73 - 6.73i)T + 89iT^{2} \)
97 \( 1 + (10.7 + 10.7i)T + 97iT^{2} \)
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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

−9.616997943340241617795904549083, −9.101994790815193070833051732215, −8.223190162436882210674367422037, −7.52615990458252025483172715804, −6.63151766833037603853695422455, −6.04765835241758243561349067601, −4.34351904927568499416815095175, −4.03006001100504638706234049863, −2.92550397726362495798756092281, −1.65311934406863986548043074393, 1.21681163303303066206761585576, 2.37913538429300117429352719027, 3.60279050954301744819823580225, 4.10731068833841924099372933064, 5.11567459048349685180308175510, 6.38008304974012167917996380471, 7.19069022784739674916353453274, 8.289738099913141736731949834080, 8.924386283432897518786346199074, 9.616605089872127032190653440365

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