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Properties

Label 2-1110-185.68-c1-0-2
Degree $2$
Conductor $1110$
Sign $-0.889 + 0.456i$
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 + (−1.17 + 1.90i)5-s + (−0.707 + 0.707i)6-s + (−2.67 + 2.67i)7-s + 8-s − 1.00i·9-s + (−1.17 + 1.90i)10-s − 2.74i·11-s + (−0.707 + 0.707i)12-s − 3.42·13-s + (−2.67 + 2.67i)14-s + (−0.510 − 2.17i)15-s + 16-s − 0.534i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.408 + 0.408i)3-s + 0.5·4-s + (−0.526 + 0.849i)5-s + (−0.288 + 0.288i)6-s + (−1.00 + 1.00i)7-s + 0.353·8-s − 0.333i·9-s + (−0.372 + 0.601i)10-s − 0.828i·11-s + (−0.204 + 0.204i)12-s − 0.951·13-s + (−0.713 + 0.713i)14-s + (−0.131 − 0.562i)15-s + 0.250·16-s − 0.129i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 + 0.456i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.889 + 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.889 + 0.456i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.889 + 0.456i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3634028768\)
\(L(\frac12)\) \(\approx\) \(0.3634028768\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.17 - 1.90i)T \)
37 \( 1 + (-0.429 + 6.06i)T \)
good7 \( 1 + (2.67 - 2.67i)T - 7iT^{2} \)
11 \( 1 + 2.74iT - 11T^{2} \)
13 \( 1 + 3.42T + 13T^{2} \)
17 \( 1 + 0.534iT - 17T^{2} \)
19 \( 1 + (1.10 + 1.10i)T + 19iT^{2} \)
23 \( 1 - 2.91T + 23T^{2} \)
29 \( 1 + (6.34 - 6.34i)T - 29iT^{2} \)
31 \( 1 + (3.92 + 3.92i)T + 31iT^{2} \)
41 \( 1 + 9.62iT - 41T^{2} \)
43 \( 1 - 1.95T + 43T^{2} \)
47 \( 1 + (9.10 - 9.10i)T - 47iT^{2} \)
53 \( 1 + (-8.83 - 8.83i)T + 53iT^{2} \)
59 \( 1 + (8.17 + 8.17i)T + 59iT^{2} \)
61 \( 1 + (-3.52 - 3.52i)T + 61iT^{2} \)
67 \( 1 + (-5.36 - 5.36i)T + 67iT^{2} \)
71 \( 1 + 15.0T + 71T^{2} \)
73 \( 1 + (7.28 - 7.28i)T - 73iT^{2} \)
79 \( 1 + (-4.60 - 4.60i)T + 79iT^{2} \)
83 \( 1 + (10.9 + 10.9i)T + 83iT^{2} \)
89 \( 1 + (9.99 - 9.99i)T - 89iT^{2} \)
97 \( 1 - 14.3iT - 97T^{2} \)
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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

−10.57880249124116414574251503590, −9.513192826128135743147461717999, −8.874423324464480780185510177818, −7.51321220768013667790691766428, −6.84927449336032504791154263391, −5.91598295658114599151157132593, −5.37705573151738587881199846341, −4.09192313745645395530790459230, −3.21722513458668707532920448072, −2.49150130429039162877610449902, 0.12464652229033167964008747751, 1.68900225782751438608478246105, 3.21882389349289663696089169159, 4.24258854085209045184648373352, 4.88852264860457344374286452554, 5.91802430764307136540651112954, 7.01114145436478331024583089457, 7.32462624248754181562689298709, 8.349098059630715249398500807825, 9.657037455230565836276780606098

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