Properties

Label 2-1110-111.68-c1-0-44
Degree $2$
Conductor $1110$
Sign $-0.628 - 0.777i$
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 + (−1.55 − 0.759i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (−1.63 + 0.564i)6-s + 1.92·7-s + (−0.707 − 0.707i)8-s + (1.84 + 2.36i)9-s + 1.00·10-s − 5.18·11-s + (−0.759 + 1.55i)12-s + (−3.96 + 3.96i)13-s + (1.36 − 1.36i)14-s + (−0.564 − 1.63i)15-s − 1.00·16-s + (−3.83 − 3.83i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.898 − 0.438i)3-s − 0.500i·4-s + (0.316 + 0.316i)5-s + (−0.668 + 0.230i)6-s + 0.727·7-s + (−0.250 − 0.250i)8-s + (0.615 + 0.787i)9-s + 0.316·10-s − 1.56·11-s + (−0.219 + 0.449i)12-s + (−1.10 + 1.10i)13-s + (0.363 − 0.363i)14-s + (−0.145 − 0.422i)15-s − 0.250·16-s + (−0.930 − 0.930i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.628 - 0.777i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.628 - 0.777i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.02323936777\)
\(L(\frac12)\) \(\approx\) \(0.02323936777\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.55 + 0.759i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (5.86 - 1.59i)T \)
good7 \( 1 - 1.92T + 7T^{2} \)
11 \( 1 + 5.18T + 11T^{2} \)
13 \( 1 + (3.96 - 3.96i)T - 13iT^{2} \)
17 \( 1 + (3.83 + 3.83i)T + 17iT^{2} \)
19 \( 1 + (-0.0989 + 0.0989i)T - 19iT^{2} \)
23 \( 1 + (1.07 + 1.07i)T + 23iT^{2} \)
29 \( 1 + (1.77 - 1.77i)T - 29iT^{2} \)
31 \( 1 + (4.45 + 4.45i)T + 31iT^{2} \)
41 \( 1 + 12.4T + 41T^{2} \)
43 \( 1 + (-7.21 + 7.21i)T - 43iT^{2} \)
47 \( 1 - 3.47iT - 47T^{2} \)
53 \( 1 + 3.95iT - 53T^{2} \)
59 \( 1 + (-7.83 - 7.83i)T + 59iT^{2} \)
61 \( 1 + (-9.10 - 9.10i)T + 61iT^{2} \)
67 \( 1 - 9.29iT - 67T^{2} \)
71 \( 1 - 1.23iT - 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 + (0.761 - 0.761i)T - 79iT^{2} \)
83 \( 1 - 9.18iT - 83T^{2} \)
89 \( 1 + (5.54 - 5.54i)T - 89iT^{2} \)
97 \( 1 + (-3.84 + 3.84i)T - 97iT^{2} \)
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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

−9.659008488321917646266191327271, −8.464087720957379589886910582697, −7.24758157691364446127166105897, −6.92776609826215772710007393653, −5.55301685582091568467434928041, −5.10840012219184625046619442154, −4.28410890132736796092439217143, −2.53434556041901627667870308788, −1.91237410272164710517859141826, −0.008751530301217760471444672518, 2.08018936118948402378859109406, 3.50716082057844731647702442537, 4.83711587971632407093240079383, 5.11946302312483160859015690279, 5.85422999969237283205874026711, 6.94138051494441678976782309322, 7.82817388889053632270590493425, 8.522660839820770857783391762922, 9.737650521146752220612796285947, 10.47771538106142668169935262905

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