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Properties

Label 2-1380-92.91-c1-0-61
Degree $2$
Conductor $1380$
Sign $0.758 + 0.652i$
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  + (−0.698 + 1.22i)2-s + i·3-s + (−1.02 − 1.71i)4-s − i·5-s + (−1.22 − 0.698i)6-s − 0.344·7-s + (2.82 − 0.0624i)8-s − 9-s + (1.22 + 0.698i)10-s − 1.11·11-s + (1.71 − 1.02i)12-s − 4.23·13-s + (0.240 − 0.423i)14-s + 15-s + (−1.89 + 3.52i)16-s + 3.31i·17-s + ⋯
L(s)  = 1  + (−0.493 + 0.869i)2-s + 0.577i·3-s + (−0.512 − 0.858i)4-s − 0.447i·5-s + (−0.502 − 0.284i)6-s − 0.130·7-s + (0.999 − 0.0220i)8-s − 0.333·9-s + (0.388 + 0.220i)10-s − 0.336·11-s + (0.495 − 0.296i)12-s − 1.17·13-s + (0.0642 − 0.113i)14-s + 0.258·15-s + (−0.474 + 0.880i)16-s + 0.804i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 + 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.758 + 0.652i$
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: \((2,\ 1380,\ (\ :1/2),\ 0.758 + 0.652i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6062723330\)
\(L(\frac12)\) \(\approx\) \(0.6062723330\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.698 - 1.22i)T \)
3 \( 1 - iT \)
5 \( 1 + iT \)
23 \( 1 + (-4.54 + 1.51i)T \)
good7 \( 1 + 0.344T + 7T^{2} \)
11 \( 1 + 1.11T + 11T^{2} \)
13 \( 1 + 4.23T + 13T^{2} \)
17 \( 1 - 3.31iT - 17T^{2} \)
19 \( 1 - 1.10T + 19T^{2} \)
29 \( 1 - 0.696T + 29T^{2} \)
31 \( 1 + 1.78iT - 31T^{2} \)
37 \( 1 + 8.80iT - 37T^{2} \)
41 \( 1 + 9.74T + 41T^{2} \)
43 \( 1 - 8.82T + 43T^{2} \)
47 \( 1 + 8.35iT - 47T^{2} \)
53 \( 1 + 12.1iT - 53T^{2} \)
59 \( 1 + 14.6iT - 59T^{2} \)
61 \( 1 - 7.06iT - 61T^{2} \)
67 \( 1 + 2.20T + 67T^{2} \)
71 \( 1 + 16.5iT - 71T^{2} \)
73 \( 1 - 2.68T + 73T^{2} \)
79 \( 1 + 15.3T + 79T^{2} \)
83 \( 1 - 17.0T + 83T^{2} \)
89 \( 1 + 11.9iT - 89T^{2} \)
97 \( 1 + 9.50iT - 97T^{2} \)
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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

−9.380179087952132637447955609580, −8.723131687729699096558724626529, −7.924951729128211118318891338116, −7.14840994500196705523691445854, −6.21000437478990336022731305828, −5.22402800157254280137032397351, −4.77174564503382279726766817406, −3.60930303604912464253948793478, −2.05656002847882945637059442729, −0.31454481428245047343530521243, 1.23404415316541756151804973527, 2.61747267487659403347376988712, 3.05734692836525106179256086608, 4.49709496986817342760694711325, 5.36029540526859177889883213046, 6.75013282897947017492101000258, 7.36765326261426069507233911543, 8.029698020116270427839570829569, 9.052508091530920270840521326028, 9.698764548961402138419873237977

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