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Properties

Label 2-1110-185.142-c1-0-37
Degree $2$
Conductor $1110$
Sign $-0.898 + 0.439i$
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 + (1.87 − 1.21i)5-s + (0.707 − 0.707i)6-s + (−2.24 − 2.24i)7-s + i·8-s + 1.00i·9-s + (−1.21 − 1.87i)10-s − 5.65i·11-s + (−0.707 − 0.707i)12-s + 4.81i·13-s + (−2.24 + 2.24i)14-s + (2.18 + 0.466i)15-s + 16-s − 7.58·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (0.839 − 0.543i)5-s + (0.288 − 0.288i)6-s + (−0.849 − 0.849i)7-s + 0.353i·8-s + 0.333i·9-s + (−0.384 − 0.593i)10-s − 1.70i·11-s + (−0.204 − 0.204i)12-s + 1.33i·13-s + (−0.600 + 0.600i)14-s + (0.564 + 0.120i)15-s + 0.250·16-s − 1.83·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 + 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.898 + 0.439i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.898 + 0.439i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.268058346\)
\(L(\frac12)\) \(\approx\) \(1.268058346\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.87 + 1.21i)T \)
37 \( 1 + (4.17 - 4.42i)T \)
good7 \( 1 + (2.24 + 2.24i)T + 7iT^{2} \)
11 \( 1 + 5.65iT - 11T^{2} \)
13 \( 1 - 4.81iT - 13T^{2} \)
17 \( 1 + 7.58T + 17T^{2} \)
19 \( 1 + (3.73 + 3.73i)T + 19iT^{2} \)
23 \( 1 + 4.23iT - 23T^{2} \)
29 \( 1 + (-3.26 + 3.26i)T - 29iT^{2} \)
31 \( 1 + (-1.98 - 1.98i)T + 31iT^{2} \)
41 \( 1 - 2.43iT - 41T^{2} \)
43 \( 1 - 3.34iT - 43T^{2} \)
47 \( 1 + (-6.04 - 6.04i)T + 47iT^{2} \)
53 \( 1 + (-8.02 + 8.02i)T - 53iT^{2} \)
59 \( 1 + (9.84 + 9.84i)T + 59iT^{2} \)
61 \( 1 + (1.11 + 1.11i)T + 61iT^{2} \)
67 \( 1 + (-6.91 + 6.91i)T - 67iT^{2} \)
71 \( 1 + 6.12T + 71T^{2} \)
73 \( 1 + (2.58 + 2.58i)T + 73iT^{2} \)
79 \( 1 + (2.85 + 2.85i)T + 79iT^{2} \)
83 \( 1 + (-0.906 + 0.906i)T - 83iT^{2} \)
89 \( 1 + (-10.1 + 10.1i)T - 89iT^{2} \)
97 \( 1 + 2.87T + 97T^{2} \)
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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

−9.404985708793926414380461137649, −8.888030745806656518699374668060, −8.375329320972583560186295995025, −6.62326980343546412060051882145, −6.31885959428174716481698152628, −4.75032641639247690980656720932, −4.22438703923597791830222622037, −3.06767691535122413135975734470, −2.09254853153426992705922227659, −0.49511322925702352762953258500, 1.99350590611982748368159116393, 2.77445175493330315301819077418, 4.11330194827207953584176088899, 5.39909902430970867450714379986, 6.09299252459573239029220720617, 6.89671770719297969613518897938, 7.45635608205045596234250320336, 8.692940665934348316444657704683, 9.173068574188824493989001944613, 10.08656890680444998625535777961

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