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Properties

Label 2-1380-92.91-c1-0-24
Degree $2$
Conductor $1380$
Sign $0.367 - 0.930i$
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  + (−1.41 + 0.0379i)2-s − i·3-s + (1.99 − 0.107i)4-s + i·5-s + (0.0379 + 1.41i)6-s + 1.92·7-s + (−2.81 + 0.227i)8-s − 9-s + (−0.0379 − 1.41i)10-s − 6.24·11-s + (−0.107 − 1.99i)12-s + 1.29·13-s + (−2.71 + 0.0729i)14-s + 15-s + (3.97 − 0.428i)16-s + 0.0974i·17-s + ⋯
L(s)  = 1  + (−0.999 + 0.0268i)2-s − 0.577i·3-s + (0.998 − 0.0536i)4-s + 0.447i·5-s + (0.0154 + 0.577i)6-s + 0.726·7-s + (−0.996 + 0.0804i)8-s − 0.333·9-s + (−0.0119 − 0.447i)10-s − 1.88·11-s + (−0.0309 − 0.576i)12-s + 0.359·13-s + (−0.726 + 0.0194i)14-s + 0.258·15-s + (0.994 − 0.107i)16-s + 0.0236i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.367 - 0.930i$
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: \((2,\ 1380,\ (\ :1/2),\ 0.367 - 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7653353994\)
\(L(\frac12)\) \(\approx\) \(0.7653353994\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0379i)T \)
3 \( 1 + iT \)
5 \( 1 - iT \)
23 \( 1 + (1.99 - 4.35i)T \)
good7 \( 1 - 1.92T + 7T^{2} \)
11 \( 1 + 6.24T + 11T^{2} \)
13 \( 1 - 1.29T + 13T^{2} \)
17 \( 1 - 0.0974iT - 17T^{2} \)
19 \( 1 - 1.71T + 19T^{2} \)
29 \( 1 - 6.14T + 29T^{2} \)
31 \( 1 - 2.62iT - 31T^{2} \)
37 \( 1 - 9.78iT - 37T^{2} \)
41 \( 1 - 9.53T + 41T^{2} \)
43 \( 1 + 4.28T + 43T^{2} \)
47 \( 1 - 4.88iT - 47T^{2} \)
53 \( 1 - 2.10iT - 53T^{2} \)
59 \( 1 + 9.68iT - 59T^{2} \)
61 \( 1 - 6.28iT - 61T^{2} \)
67 \( 1 + 4.41T + 67T^{2} \)
71 \( 1 - 10.2iT - 71T^{2} \)
73 \( 1 + 6.93T + 73T^{2} \)
79 \( 1 - 7.77T + 79T^{2} \)
83 \( 1 + 2.75T + 83T^{2} \)
89 \( 1 - 7.14iT - 89T^{2} \)
97 \( 1 - 6.15iT - 97T^{2} \)
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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

−9.854771094664119218598660728343, −8.713038718976304048154525075915, −7.904892168017864916645100617428, −7.70419443081746268083884572986, −6.67612752539147526352761835945, −5.80539582640940460731630354094, −4.92898182786936993697526834680, −3.17278373580464035572922322217, −2.40148484360596121647111286322, −1.22944005723318418137225333554, 0.45640123507406395479592175278, 2.06369652751200531869040987187, 2.99968338299999838971336629317, 4.40021600652822520914641206263, 5.31181543303632070873589911278, 6.06014621799575060954290463062, 7.37504747889309893919363046463, 8.043112998503057967626008484264, 8.552128163783387940305595492560, 9.410788655345689556012770504021

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