Properties

Label 2-1110-185.142-c1-0-18
Degree $2$
Conductor $1110$
Sign $0.981 + 0.190i$
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  i·2-s + (0.707 + 0.707i)3-s − 4-s + (1.45 − 1.69i)5-s + (0.707 − 0.707i)6-s + (−0.764 − 0.764i)7-s + i·8-s + 1.00i·9-s + (−1.69 − 1.45i)10-s + 4.86i·11-s + (−0.707 − 0.707i)12-s + 4.63i·13-s + (−0.764 + 0.764i)14-s + (2.22 − 0.170i)15-s + 16-s + 5.18·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (0.650 − 0.759i)5-s + (0.288 − 0.288i)6-s + (−0.288 − 0.288i)7-s + 0.353i·8-s + 0.333i·9-s + (−0.536 − 0.460i)10-s + 1.46i·11-s + (−0.204 − 0.204i)12-s + 1.28i·13-s + (−0.204 + 0.204i)14-s + (0.575 − 0.0441i)15-s + 0.250·16-s + 1.25·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.190i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.981 + 0.190i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.981 + 0.190i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.973668954\)
\(L(\frac12)\) \(\approx\) \(1.973668954\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-1.45 + 1.69i)T \)
37 \( 1 + (-6.03 - 0.720i)T \)
good7 \( 1 + (0.764 + 0.764i)T + 7iT^{2} \)
11 \( 1 - 4.86iT - 11T^{2} \)
13 \( 1 - 4.63iT - 13T^{2} \)
17 \( 1 - 5.18T + 17T^{2} \)
19 \( 1 + (-1.84 - 1.84i)T + 19iT^{2} \)
23 \( 1 - 4.72iT - 23T^{2} \)
29 \( 1 + (-6.46 + 6.46i)T - 29iT^{2} \)
31 \( 1 + (-1.27 - 1.27i)T + 31iT^{2} \)
41 \( 1 - 3.55iT - 41T^{2} \)
43 \( 1 + 7.49iT - 43T^{2} \)
47 \( 1 + (8.00 + 8.00i)T + 47iT^{2} \)
53 \( 1 + (5.64 - 5.64i)T - 53iT^{2} \)
59 \( 1 + (-0.790 - 0.790i)T + 59iT^{2} \)
61 \( 1 + (5.24 + 5.24i)T + 61iT^{2} \)
67 \( 1 + (-0.777 + 0.777i)T - 67iT^{2} \)
71 \( 1 - 0.599T + 71T^{2} \)
73 \( 1 + (-1.07 - 1.07i)T + 73iT^{2} \)
79 \( 1 + (-9.56 - 9.56i)T + 79iT^{2} \)
83 \( 1 + (11.2 - 11.2i)T - 83iT^{2} \)
89 \( 1 + (-1.70 + 1.70i)T - 89iT^{2} \)
97 \( 1 - 6.03T + 97T^{2} \)
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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

−9.812787967499779462788819855224, −9.418070474046680090233733511611, −8.380513518728621799826814476539, −7.52259454960391635247344419576, −6.38953807654039115235169146367, −5.19918296551572773153590458897, −4.50359398676062240679957417071, −3.63561456347231234481501218944, −2.30121751573834526166964323485, −1.37440794059554517095115735639, 0.969981551396014842670380178087, 2.97889115555505894086193557552, 3.16472603087485516281901867369, 4.98527665207218283931540062607, 6.02369370746777568933416883199, 6.27039155377064439872191973784, 7.46091802300649510517992778773, 8.103468012330117537051001233872, 8.883375141169657597938094139761, 9.780017646032544096907406775539

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