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Properties

Label 2-1110-111.68-c1-0-26
Degree $2$
Conductor $1110$
Sign $0.137 + 0.990i$
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 + (−0.878 − 1.49i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (−1.67 − 0.434i)6-s + 2.38·7-s + (−0.707 − 0.707i)8-s + (−1.45 + 2.62i)9-s + 1.00·10-s + 5.76·11-s + (−1.49 + 0.878i)12-s + (−1.58 + 1.58i)13-s + (1.68 − 1.68i)14-s + (0.434 − 1.67i)15-s − 1.00·16-s + (−0.766 − 0.766i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.507 − 0.861i)3-s − 0.500i·4-s + (0.316 + 0.316i)5-s + (−0.684 − 0.177i)6-s + 0.901·7-s + (−0.250 − 0.250i)8-s + (−0.485 + 0.874i)9-s + 0.316·10-s + 1.73·11-s + (−0.430 + 0.253i)12-s + (−0.438 + 0.438i)13-s + (0.450 − 0.450i)14-s + (0.112 − 0.432i)15-s − 0.250·16-s + (−0.186 − 0.186i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.137 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.137 + 0.990i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.137 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.251244665\)
\(L(\frac12)\) \(\approx\) \(2.251244665\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.878 + 1.49i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (-5.79 + 1.85i)T \)
good7 \( 1 - 2.38T + 7T^{2} \)
11 \( 1 - 5.76T + 11T^{2} \)
13 \( 1 + (1.58 - 1.58i)T - 13iT^{2} \)
17 \( 1 + (0.766 + 0.766i)T + 17iT^{2} \)
19 \( 1 + (-4.27 + 4.27i)T - 19iT^{2} \)
23 \( 1 + (-5.87 - 5.87i)T + 23iT^{2} \)
29 \( 1 + (0.666 - 0.666i)T - 29iT^{2} \)
31 \( 1 + (4.93 + 4.93i)T + 31iT^{2} \)
41 \( 1 - 2.85T + 41T^{2} \)
43 \( 1 + (6.47 - 6.47i)T - 43iT^{2} \)
47 \( 1 + 8.44iT - 47T^{2} \)
53 \( 1 - 7.21iT - 53T^{2} \)
59 \( 1 + (5.63 + 5.63i)T + 59iT^{2} \)
61 \( 1 + (8.82 + 8.82i)T + 61iT^{2} \)
67 \( 1 + 2.96iT - 67T^{2} \)
71 \( 1 + 10.2iT - 71T^{2} \)
73 \( 1 + 1.48iT - 73T^{2} \)
79 \( 1 + (3.85 - 3.85i)T - 79iT^{2} \)
83 \( 1 - 16.2iT - 83T^{2} \)
89 \( 1 + (3.88 - 3.88i)T - 89iT^{2} \)
97 \( 1 + (-11.4 + 11.4i)T - 97iT^{2} \)
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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

−9.509855299766313439394795011562, −9.146090770572154553658984430013, −7.72776517247304260177331057013, −7.02191565970408603800892065228, −6.29545758938073367661481643180, −5.30009888549124847938070022903, −4.57587386504736256664593042321, −3.25574239566084489713644307416, −1.97927980149927188212505988401, −1.16045647555273202155420515377, 1.32370477216399643352582182071, 3.15484065276119191299694546095, 4.20617949463455724247839597277, 4.84555793291087919636442161778, 5.66493533545734920544488519840, 6.44846572859651642943403263278, 7.40168632560391939103111554401, 8.572938135818179945124100224344, 9.099758173060531467441825255019, 9.994080657110618551293996763987

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