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Properties

Label 2-1110-185.68-c1-0-1
Degree $2$
Conductor $1110$
Sign $-0.444 + 0.895i$
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  − 2-s + (−0.707 + 0.707i)3-s + 4-s + (−2.22 − 0.262i)5-s + (0.707 − 0.707i)6-s + (−2.69 + 2.69i)7-s − 8-s − 1.00i·9-s + (2.22 + 0.262i)10-s + 3.85i·11-s + (−0.707 + 0.707i)12-s + 1.07·13-s + (2.69 − 2.69i)14-s + (1.75 − 1.38i)15-s + 16-s + 3.12i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.408 + 0.408i)3-s + 0.5·4-s + (−0.993 − 0.117i)5-s + (0.288 − 0.288i)6-s + (−1.01 + 1.01i)7-s − 0.353·8-s − 0.333i·9-s + (0.702 + 0.0829i)10-s + 1.16i·11-s + (−0.204 + 0.204i)12-s + 0.297·13-s + (0.720 − 0.720i)14-s + (0.453 − 0.357i)15-s + 0.250·16-s + 0.758i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.444 + 0.895i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.444 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.06637717628\)
\(L(\frac12)\) \(\approx\) \(0.06637717628\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (2.22 + 0.262i)T \)
37 \( 1 + (-6.03 + 0.778i)T \)
good7 \( 1 + (2.69 - 2.69i)T - 7iT^{2} \)
11 \( 1 - 3.85iT - 11T^{2} \)
13 \( 1 - 1.07T + 13T^{2} \)
17 \( 1 - 3.12iT - 17T^{2} \)
19 \( 1 + (-4.87 - 4.87i)T + 19iT^{2} \)
23 \( 1 + 6.72T + 23T^{2} \)
29 \( 1 + (0.240 - 0.240i)T - 29iT^{2} \)
31 \( 1 + (7.29 + 7.29i)T + 31iT^{2} \)
41 \( 1 + 9.24iT - 41T^{2} \)
43 \( 1 + 8.98T + 43T^{2} \)
47 \( 1 + (3.13 - 3.13i)T - 47iT^{2} \)
53 \( 1 + (6.75 + 6.75i)T + 53iT^{2} \)
59 \( 1 + (-1.03 - 1.03i)T + 59iT^{2} \)
61 \( 1 + (-9.33 - 9.33i)T + 61iT^{2} \)
67 \( 1 + (-0.0813 - 0.0813i)T + 67iT^{2} \)
71 \( 1 + 9.73T + 71T^{2} \)
73 \( 1 + (-3.77 + 3.77i)T - 73iT^{2} \)
79 \( 1 + (-4.33 - 4.33i)T + 79iT^{2} \)
83 \( 1 + (5.06 + 5.06i)T + 83iT^{2} \)
89 \( 1 + (-4.92 + 4.92i)T - 89iT^{2} \)
97 \( 1 - 2.13iT - 97T^{2} \)
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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

−10.05038676826392542220442711973, −9.748748295739150195384986015633, −8.843265024094727766273816518035, −7.983402659897735648641388482229, −7.23528272363352276825633970946, −6.17524193078042212043085609824, −5.51513931879872873530947318652, −4.12783234438355608895653986242, −3.36675320460933002359564198860, −1.92654899351432119153713463625, 0.04868863157878892607114418274, 0.991873592833105708046636267766, 3.00923820490147493366566908430, 3.65434139887825347452703261378, 5.01874907319368129834215539721, 6.28529101617620467766938875346, 6.89209006243988410046791646808, 7.60666361799786236595365234289, 8.319056332727795787490776716704, 9.333304597133387899840972558121

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