Properties

Label 2-1110-111.68-c1-0-8
Degree $2$
Conductor $1110$
Sign $-0.996 + 0.0872i$
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 + (−1.70 + 0.292i)6-s − 2·7-s + (0.707 + 0.707i)8-s + (1.00 + 2.82i)9-s − 1.00·10-s − 2.82·11-s + (1.00 − 1.41i)12-s + (−3 + 3i)13-s + (1.41 − 1.41i)14-s + (0.292 + 1.70i)15-s − 1.00·16-s + (1.41 + 1.41i)17-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 + (−0.696 + 0.119i)6-s − 0.755·7-s + (0.250 + 0.250i)8-s + (0.333 + 0.942i)9-s − 0.316·10-s − 0.852·11-s + (0.288 − 0.408i)12-s + (−0.832 + 0.832i)13-s + (0.377 − 0.377i)14-s + (0.0756 + 0.440i)15-s − 0.250·16-s + (0.342 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.996 + 0.0872i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.996 + 0.0872i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8672846867\)
\(L(\frac12)\) \(\approx\) \(0.8672846867\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 + (3 - 3i)T - 13iT^{2} \)
17 \( 1 + (-1.41 - 1.41i)T + 17iT^{2} \)
19 \( 1 + (1 - i)T - 19iT^{2} \)
23 \( 1 + (5.65 + 5.65i)T + 23iT^{2} \)
29 \( 1 + (-1.41 + 1.41i)T - 29iT^{2} \)
31 \( 1 + (-1 - i)T + 31iT^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + (5 - 5i)T - 43iT^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 5.65iT - 53T^{2} \)
59 \( 1 + (2.82 + 2.82i)T + 59iT^{2} \)
61 \( 1 + (-9 - 9i)T + 61iT^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + (-3 + 3i)T - 79iT^{2} \)
83 \( 1 - 2.82iT - 83T^{2} \)
89 \( 1 + (-7.07 + 7.07i)T - 89iT^{2} \)
97 \( 1 + (3 - 3i)T - 97iT^{2} \)
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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

−10.01817937744946976879595112319, −9.614796113478722431272988127242, −8.532626322373114261609186256585, −8.007844563322058477479786869695, −6.99698298852707931511262404097, −6.25628753844529620627550210921, −5.12934463129489083380854484738, −4.18944925528179943722425542347, −2.94120200242710195469088608689, −2.05120581555014585016485864331, 0.37185346704199956613551381823, 1.94255174828790417541021559545, 2.84015135250746152488642334774, 3.66849998151823436401108988228, 5.10748749137227805131380052097, 6.16153169140429698541958761165, 7.29048148965122566274021782540, 7.82718986756229748530228392172, 8.626739136145718912353280464118, 9.632781547312920711115918691044

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