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Properties

Label 2-1110-185.68-c1-0-12
Degree $2$
Conductor $1110$
Sign $0.316 - 0.948i$
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 + (0.837 + 2.07i)5-s + (0.707 − 0.707i)6-s + (−2.93 + 2.93i)7-s + 8-s − 1.00i·9-s + (0.837 + 2.07i)10-s + 1.12i·11-s + (0.707 − 0.707i)12-s − 2.10·13-s + (−2.93 + 2.93i)14-s + (2.05 + 0.874i)15-s + 16-s + 3.81i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.408 − 0.408i)3-s + 0.5·4-s + (0.374 + 0.927i)5-s + (0.288 − 0.288i)6-s + (−1.11 + 1.11i)7-s + 0.353·8-s − 0.333i·9-s + (0.264 + 0.655i)10-s + 0.340i·11-s + (0.204 − 0.204i)12-s − 0.583·13-s + (−0.785 + 0.785i)14-s + (0.531 + 0.225i)15-s + 0.250·16-s + 0.926i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.316 - 0.948i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.316 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.526142822\)
\(L(\frac12)\) \(\approx\) \(2.526142822\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.837 - 2.07i)T \)
37 \( 1 + (-1.41 + 5.91i)T \)
good7 \( 1 + (2.93 - 2.93i)T - 7iT^{2} \)
11 \( 1 - 1.12iT - 11T^{2} \)
13 \( 1 + 2.10T + 13T^{2} \)
17 \( 1 - 3.81iT - 17T^{2} \)
19 \( 1 + (-0.0424 - 0.0424i)T + 19iT^{2} \)
23 \( 1 - 7.44T + 23T^{2} \)
29 \( 1 + (1.00 - 1.00i)T - 29iT^{2} \)
31 \( 1 + (-5.55 - 5.55i)T + 31iT^{2} \)
41 \( 1 - 7.31iT - 41T^{2} \)
43 \( 1 + 6.59T + 43T^{2} \)
47 \( 1 + (-3.52 + 3.52i)T - 47iT^{2} \)
53 \( 1 + (6.43 + 6.43i)T + 53iT^{2} \)
59 \( 1 + (1.53 + 1.53i)T + 59iT^{2} \)
61 \( 1 + (-3.13 - 3.13i)T + 61iT^{2} \)
67 \( 1 + (2.59 + 2.59i)T + 67iT^{2} \)
71 \( 1 - 2.10T + 71T^{2} \)
73 \( 1 + (-6.95 + 6.95i)T - 73iT^{2} \)
79 \( 1 + (-8.44 - 8.44i)T + 79iT^{2} \)
83 \( 1 + (2.80 + 2.80i)T + 83iT^{2} \)
89 \( 1 + (-8.09 + 8.09i)T - 89iT^{2} \)
97 \( 1 + 10.4iT - 97T^{2} \)
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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

−9.954921136807405897593407052308, −9.302517891529294347372104461357, −8.329332871974412881995596124157, −7.17319521272080005559157038155, −6.60087816616640576570655646128, −5.94264978203010107438848722709, −4.94054710574855245225137833660, −3.42966320215986259501894149190, −2.87270346918100019517379408208, −1.95837926567475059175478728797, 0.826751940236954873091064049781, 2.59133365715819793858893629422, 3.50111625728289596213576687990, 4.48174282958553040213176876450, 5.13230201134933038920040799388, 6.25214825275272322365804166532, 7.08813464792113942695736627899, 7.946123760607607027070620169910, 9.100523705945599282912452670818, 9.669845236194026184645850848315

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