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Properties

Label 2-1380-345.344-c1-0-15
Degree $2$
Conductor $1380$
Sign $0.634 - 0.773i$
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  + (−0.888 + 1.48i)3-s + (−1.29 − 1.82i)5-s + 0.470·7-s + (−1.42 − 2.64i)9-s − 2.71·11-s + 2.59i·13-s + (3.86 − 0.313i)15-s − 0.294i·17-s − 0.759i·19-s + (−0.417 + 0.699i)21-s + (3.33 − 3.44i)23-s + (−1.62 + 4.72i)25-s + (5.19 + 0.229i)27-s + 6.71i·29-s + 2.75·31-s + ⋯
L(s)  = 1  + (−0.512 + 0.858i)3-s + (−0.580 − 0.814i)5-s + 0.177·7-s + (−0.474 − 0.880i)9-s − 0.819·11-s + 0.719i·13-s + (0.996 − 0.0809i)15-s − 0.0713i·17-s − 0.174i·19-s + (−0.0911 + 0.152i)21-s + (0.694 − 0.719i)23-s + (−0.325 + 0.945i)25-s + (0.999 + 0.0441i)27-s + 1.24i·29-s + 0.494·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 - 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.634 - 0.773i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ 0.634 - 0.773i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.037161086\)
\(L(\frac12)\) \(\approx\) \(1.037161086\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.888 - 1.48i)T \)
5 \( 1 + (1.29 + 1.82i)T \)
23 \( 1 + (-3.33 + 3.44i)T \)
good7 \( 1 - 0.470T + 7T^{2} \)
11 \( 1 + 2.71T + 11T^{2} \)
13 \( 1 - 2.59iT - 13T^{2} \)
17 \( 1 + 0.294iT - 17T^{2} \)
19 \( 1 + 0.759iT - 19T^{2} \)
29 \( 1 - 6.71iT - 29T^{2} \)
31 \( 1 - 2.75T + 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 - 1.64iT - 41T^{2} \)
43 \( 1 - 3.08T + 43T^{2} \)
47 \( 1 - 9.53T + 47T^{2} \)
53 \( 1 - 2.10iT - 53T^{2} \)
59 \( 1 + 5.16iT - 59T^{2} \)
61 \( 1 - 8.99iT - 61T^{2} \)
67 \( 1 - 9.92T + 67T^{2} \)
71 \( 1 - 2.98iT - 71T^{2} \)
73 \( 1 - 8.50iT - 73T^{2} \)
79 \( 1 + 14.8iT - 79T^{2} \)
83 \( 1 - 10.6iT - 83T^{2} \)
89 \( 1 - 8.27T + 89T^{2} \)
97 \( 1 + 0.506T + 97T^{2} \)
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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

−9.600796479057774495735338724058, −8.973910151525006433164098057071, −8.260254313440861983454115311887, −7.30033411337928284805552135070, −6.27426974524229367284694145528, −5.24114737707436848391495984468, −4.69067124005889533064658392058, −3.92699701902755307124450665241, −2.71579015874533623633890133434, −0.895474201234735005558654642591, 0.62983573939030341761016052722, 2.27781377055076101924277793975, 3.11234576394364490207175381450, 4.42266438151017237709597087386, 5.50149340115704998404080146095, 6.19711505939084802619934804319, 7.12905432975644019259731015774, 7.84414199017494891335037325708, 8.180076071607208185783835712919, 9.594914456065704402180674255566

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