Properties

Label 2-1110-111.68-c1-0-23
Degree $2$
Conductor $1110$
Sign $0.993 - 0.111i$
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 + (−0.349 + 1.69i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (0.952 + 1.44i)6-s + 2.97·7-s + (−0.707 − 0.707i)8-s + (−2.75 − 1.18i)9-s + 1.00·10-s − 2.43·11-s + (1.69 + 0.349i)12-s + (3.37 − 3.37i)13-s + (2.10 − 2.10i)14-s + (−1.44 + 0.952i)15-s − 1.00·16-s + (2.64 + 2.64i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.201 + 0.979i)3-s − 0.500i·4-s + (0.316 + 0.316i)5-s + (0.388 + 0.590i)6-s + 1.12·7-s + (−0.250 − 0.250i)8-s + (−0.918 − 0.395i)9-s + 0.316·10-s − 0.735·11-s + (0.489 + 0.100i)12-s + (0.937 − 0.937i)13-s + (0.562 − 0.562i)14-s + (−0.373 + 0.245i)15-s − 0.250·16-s + (0.640 + 0.640i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.993 - 0.111i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.993 - 0.111i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.372079594\)
\(L(\frac12)\) \(\approx\) \(2.372079594\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.349 - 1.69i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (-5.17 - 3.19i)T \)
good7 \( 1 - 2.97T + 7T^{2} \)
11 \( 1 + 2.43T + 11T^{2} \)
13 \( 1 + (-3.37 + 3.37i)T - 13iT^{2} \)
17 \( 1 + (-2.64 - 2.64i)T + 17iT^{2} \)
19 \( 1 + (-5.18 + 5.18i)T - 19iT^{2} \)
23 \( 1 + (-4.56 - 4.56i)T + 23iT^{2} \)
29 \( 1 + (2.17 - 2.17i)T - 29iT^{2} \)
31 \( 1 + (1.20 + 1.20i)T + 31iT^{2} \)
41 \( 1 + 7.04T + 41T^{2} \)
43 \( 1 + (3.83 - 3.83i)T - 43iT^{2} \)
47 \( 1 - 6.56iT - 47T^{2} \)
53 \( 1 - 0.00856iT - 53T^{2} \)
59 \( 1 + (-3.57 - 3.57i)T + 59iT^{2} \)
61 \( 1 + (-8.28 - 8.28i)T + 61iT^{2} \)
67 \( 1 - 3.64iT - 67T^{2} \)
71 \( 1 + 9.63iT - 71T^{2} \)
73 \( 1 + 5.75iT - 73T^{2} \)
79 \( 1 + (-6.34 + 6.34i)T - 79iT^{2} \)
83 \( 1 + 16.7iT - 83T^{2} \)
89 \( 1 + (-1.72 + 1.72i)T - 89iT^{2} \)
97 \( 1 + (4.30 - 4.30i)T - 97iT^{2} \)
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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

−10.09291470181718321690494610462, −9.261286987591339957630662492048, −8.336964915384092175099964931809, −7.45317320116182163999527294293, −6.04390126872269886363025170689, −5.29816578588310987533648911783, −4.83403045770655277600183966846, −3.51824914703894107216010870768, −2.87292903871650295924238537800, −1.24990556964911917802379451008, 1.20661509247155383175520678071, 2.31042709735400199509680647839, 3.68763574246663543830229668010, 5.12278776556729894737674461856, 5.38970128882798277105219433424, 6.49382886611279238107988175079, 7.28929799064365432447424604635, 8.117499939136306126339669481137, 8.554733301819541868683051098829, 9.728201815845194563939638810110

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