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Properties

Label 2-1380-92.91-c1-0-65
Degree $2$
Conductor $1380$
Sign $-0.570 + 0.821i$
Analytic cond. $11.0193$
Root an. cond. $3.31954$
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  + (0.449 − 1.34i)2-s − i·3-s + (−1.59 − 1.20i)4-s + i·5-s + (−1.34 − 0.449i)6-s + 0.628·7-s + (−2.33 + 1.59i)8-s − 9-s + (1.34 + 0.449i)10-s + 5.39·11-s + (−1.20 + 1.59i)12-s + 2.49·13-s + (0.282 − 0.842i)14-s + 15-s + (1.09 + 3.84i)16-s − 6.82i·17-s + ⋯
L(s)  = 1  + (0.317 − 0.948i)2-s − 0.577i·3-s + (−0.797 − 0.602i)4-s + 0.447i·5-s + (−0.547 − 0.183i)6-s + 0.237·7-s + (−0.825 + 0.564i)8-s − 0.333·9-s + (0.424 + 0.142i)10-s + 1.62·11-s + (−0.348 + 0.460i)12-s + 0.693·13-s + (0.0754 − 0.225i)14-s + 0.258·15-s + (0.273 + 0.961i)16-s − 1.65i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.570 + 0.821i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ -0.570 + 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.987400784\)
\(L(\frac12)\) \(\approx\) \(1.987400784\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.449 + 1.34i)T \)
3 \( 1 + iT \)
5 \( 1 - iT \)
23 \( 1 + (-0.188 - 4.79i)T \)
good7 \( 1 - 0.628T + 7T^{2} \)
11 \( 1 - 5.39T + 11T^{2} \)
13 \( 1 - 2.49T + 13T^{2} \)
17 \( 1 + 6.82iT - 17T^{2} \)
19 \( 1 - 2.30T + 19T^{2} \)
29 \( 1 - 0.986T + 29T^{2} \)
31 \( 1 + 6.92iT - 31T^{2} \)
37 \( 1 - 4.11iT - 37T^{2} \)
41 \( 1 + 0.870T + 41T^{2} \)
43 \( 1 - 4.75T + 43T^{2} \)
47 \( 1 + 12.0iT - 47T^{2} \)
53 \( 1 + 9.15iT - 53T^{2} \)
59 \( 1 - 13.4iT - 59T^{2} \)
61 \( 1 + 11.4iT - 61T^{2} \)
67 \( 1 + 2.86T + 67T^{2} \)
71 \( 1 + 0.0575iT - 71T^{2} \)
73 \( 1 + 15.9T + 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 + 9.99T + 83T^{2} \)
89 \( 1 - 0.244iT - 89T^{2} \)
97 \( 1 - 10.2iT - 97T^{2} \)
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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

−9.384386286509644364698117051892, −8.769904875922516462396029902303, −7.65556504817479488229607733276, −6.75591663136688355728652543664, −5.95955798758308248216924809834, −4.99215065833012118273537512898, −3.87798154525528803271166078844, −3.11655116371977157907210813274, −1.92185994610251039307711432184, −0.887289747653072679068501920395, 1.31355572318482989640873882339, 3.30976669810111168336117561902, 4.12878584868834246852811904522, 4.69348024734790378979345425886, 5.93941414202509106864843941089, 6.29517300352843867731079762666, 7.39554542303645663116157443039, 8.483194365810202977442058457585, 8.780132608416482604938373564084, 9.559379944487266085397521895583

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