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Properties

Label 2-1110-111.68-c1-0-31
Degree $2$
Conductor $1110$
Sign $-0.249 + 0.968i$
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 + (−1.41 + i)3-s − 1.00i·4-s + (−0.707 − 0.707i)5-s + (0.292 − 1.70i)6-s + 2·7-s + (0.707 + 0.707i)8-s + (1.00 − 2.82i)9-s + 1.00·10-s + (1.00 + 1.41i)12-s + (1 − i)13-s + (−1.41 + 1.41i)14-s + (1.70 + 0.292i)15-s − 1.00·16-s + (−4.24 − 4.24i)17-s + (1.29 + 2.70i)18-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.816 + 0.577i)3-s − 0.500i·4-s + (−0.316 − 0.316i)5-s + (0.119 − 0.696i)6-s + 0.755·7-s + (0.250 + 0.250i)8-s + (0.333 − 0.942i)9-s + 0.316·10-s + (0.288 + 0.408i)12-s + (0.277 − 0.277i)13-s + (−0.377 + 0.377i)14-s + (0.440 + 0.0756i)15-s − 0.250·16-s + (−1.02 − 1.02i)17-s + (0.304 + 0.638i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.249 + 0.968i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.249 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2631128717\)
\(L(\frac12)\) \(\approx\) \(0.2631128717\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 - i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (1 - 6i)T \)
good7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-1 + i)T - 13iT^{2} \)
17 \( 1 + (4.24 + 4.24i)T + 17iT^{2} \)
19 \( 1 + (3 - 3i)T - 19iT^{2} \)
23 \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \)
29 \( 1 + (4.24 - 4.24i)T - 29iT^{2} \)
31 \( 1 + (7 + 7i)T + 31iT^{2} \)
41 \( 1 + 5.65T + 41T^{2} \)
43 \( 1 + (-7 + 7i)T - 43iT^{2} \)
47 \( 1 - 2.82iT - 47T^{2} \)
53 \( 1 + 5.65iT - 53T^{2} \)
59 \( 1 + (5.65 + 5.65i)T + 59iT^{2} \)
61 \( 1 + (9 + 9i)T + 61iT^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 2.82iT - 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + (-3 + 3i)T - 79iT^{2} \)
83 \( 1 + 2.82iT - 83T^{2} \)
89 \( 1 + (-1.41 + 1.41i)T - 89iT^{2} \)
97 \( 1 + (-7 + 7i)T - 97iT^{2} \)
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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

−9.411344684351459884532897647738, −8.947831170893785557855074771772, −7.932072474015699313695133216875, −7.14135111731970326643039348502, −6.19995369563461895392941089814, −5.26837913012982876458972703885, −4.68132313320446542678656033379, −3.60068685911881274103624927916, −1.69605837434562825066105169149, −0.15665999798825613374994193693, 1.47071880303995430767020659839, 2.42410431095304998578483844774, 4.00067505904214859449536790195, 4.81085766092619711605133367371, 6.02214866294072957680886457212, 6.86657709383484072232327963276, 7.58544964106855497942900675124, 8.476743266371307245838190761855, 9.138221323120998059259512087705, 10.57250426468755360135818824366

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