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Properties

Label 2-1110-111.68-c1-0-21
Degree $2$
Conductor $1110$
Sign $0.991 + 0.127i$
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 + (0.392 + 1.68i)3-s − 1.00i·4-s + (−0.707 − 0.707i)5-s + (1.47 + 0.915i)6-s − 0.873·7-s + (−0.707 − 0.707i)8-s + (−2.69 + 1.32i)9-s − 1.00·10-s + 3.24·11-s + (1.68 − 0.392i)12-s + (2.80 − 2.80i)13-s + (−0.617 + 0.617i)14-s + (0.915 − 1.47i)15-s − 1.00·16-s + (2.67 + 2.67i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.226 + 0.973i)3-s − 0.500i·4-s + (−0.316 − 0.316i)5-s + (0.600 + 0.373i)6-s − 0.330·7-s + (−0.250 − 0.250i)8-s + (−0.897 + 0.441i)9-s − 0.316·10-s + 0.978·11-s + (0.486 − 0.113i)12-s + (0.776 − 0.776i)13-s + (−0.165 + 0.165i)14-s + (0.236 − 0.379i)15-s − 0.250·16-s + (0.649 + 0.649i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.991 + 0.127i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.991 + 0.127i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.252554729\)
\(L(\frac12)\) \(\approx\) \(2.252554729\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.392 - 1.68i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (2.06 + 5.72i)T \)
good7 \( 1 + 0.873T + 7T^{2} \)
11 \( 1 - 3.24T + 11T^{2} \)
13 \( 1 + (-2.80 + 2.80i)T - 13iT^{2} \)
17 \( 1 + (-2.67 - 2.67i)T + 17iT^{2} \)
19 \( 1 + (-2.24 + 2.24i)T - 19iT^{2} \)
23 \( 1 + (-3.93 - 3.93i)T + 23iT^{2} \)
29 \( 1 + (-0.0117 + 0.0117i)T - 29iT^{2} \)
31 \( 1 + (-5.20 - 5.20i)T + 31iT^{2} \)
41 \( 1 - 12.1T + 41T^{2} \)
43 \( 1 + (-8.06 + 8.06i)T - 43iT^{2} \)
47 \( 1 - 3.06iT - 47T^{2} \)
53 \( 1 - 9.17iT - 53T^{2} \)
59 \( 1 + (7.96 + 7.96i)T + 59iT^{2} \)
61 \( 1 + (6.64 + 6.64i)T + 61iT^{2} \)
67 \( 1 - 7.74iT - 67T^{2} \)
71 \( 1 + 0.580iT - 71T^{2} \)
73 \( 1 - 11.0iT - 73T^{2} \)
79 \( 1 + (-0.763 + 0.763i)T - 79iT^{2} \)
83 \( 1 - 6.26iT - 83T^{2} \)
89 \( 1 + (6.49 - 6.49i)T - 89iT^{2} \)
97 \( 1 + (-7.15 + 7.15i)T - 97iT^{2} \)
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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

−9.778161498856334263205840856490, −9.209744672058668004935269241754, −8.463697968949228662201133927016, −7.39358222002664203924890470775, −6.08663255583863827068300789566, −5.42181296777355195663913094318, −4.37820945629834798130879436455, −3.62241446025606545121224978342, −2.91777926138285916989262454181, −1.13078955215170681738498497593, 1.14566532097837593337470307863, 2.71272222572808106356918129923, 3.58198296945375338437218209555, 4.62354612945363131087870689917, 6.08891023694200245294497147638, 6.38157312629727924867424106237, 7.30751729583009729320098947989, 7.941653470825992488262294384695, 8.909300732892241261025464161822, 9.548290116091531981921117902632

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