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Properties

Label 2-1380-115.68-c1-0-8
Degree $2$
Conductor $1380$
Sign $0.506 + 0.862i$
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  + (−0.707 − 0.707i)3-s + (−1.68 + 1.47i)5-s + (−2.29 − 2.29i)7-s + 1.00i·9-s + 5.35i·11-s + (−2.00 − 2.00i)13-s + (2.23 + 0.151i)15-s + (3.44 + 3.44i)17-s − 0.00250·19-s + 3.24i·21-s + (1.94 − 4.38i)23-s + (0.676 − 4.95i)25-s + (0.707 − 0.707i)27-s − 1.32i·29-s + 7.62·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.753 + 0.657i)5-s + (−0.867 − 0.867i)7-s + 0.333i·9-s + 1.61i·11-s + (−0.556 − 0.556i)13-s + (0.576 + 0.0391i)15-s + (0.836 + 0.836i)17-s − 0.000575·19-s + 0.708i·21-s + (0.405 − 0.913i)23-s + (0.135 − 0.990i)25-s + (0.136 − 0.136i)27-s − 0.245i·29-s + 1.36·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.506 + 0.862i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (1333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ 0.506 + 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8363928440\)
\(L(\frac12)\) \(\approx\) \(0.8363928440\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (1.68 - 1.47i)T \)
23 \( 1 + (-1.94 + 4.38i)T \)
good7 \( 1 + (2.29 + 2.29i)T + 7iT^{2} \)
11 \( 1 - 5.35iT - 11T^{2} \)
13 \( 1 + (2.00 + 2.00i)T + 13iT^{2} \)
17 \( 1 + (-3.44 - 3.44i)T + 17iT^{2} \)
19 \( 1 + 0.00250T + 19T^{2} \)
29 \( 1 + 1.32iT - 29T^{2} \)
31 \( 1 - 7.62T + 31T^{2} \)
37 \( 1 + (6.26 + 6.26i)T + 37iT^{2} \)
41 \( 1 + 7.53T + 41T^{2} \)
43 \( 1 + (0.265 - 0.265i)T - 43iT^{2} \)
47 \( 1 + (-8.13 + 8.13i)T - 47iT^{2} \)
53 \( 1 + (-2.78 + 2.78i)T - 53iT^{2} \)
59 \( 1 + 10.7iT - 59T^{2} \)
61 \( 1 - 2.04iT - 61T^{2} \)
67 \( 1 + (-2.91 - 2.91i)T + 67iT^{2} \)
71 \( 1 - 7.55T + 71T^{2} \)
73 \( 1 + (0.944 + 0.944i)T + 73iT^{2} \)
79 \( 1 - 6.43T + 79T^{2} \)
83 \( 1 + (2.53 - 2.53i)T - 83iT^{2} \)
89 \( 1 - 8.05T + 89T^{2} \)
97 \( 1 + (-9.47 - 9.47i)T + 97iT^{2} \)
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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

−9.847335634677455447505039283172, −8.400577197717245187453556424225, −7.59866070208531686798147858181, −6.96386601993070249749121051705, −6.53000371868900111088132698051, −5.22821757436585555209496679917, −4.23881631037320120154009675256, −3.39646403374257220296057051951, −2.20611047147616104898703272819, −0.48847942811423706240996430916, 0.909248122313574789243388881615, 2.93234565208010389776447514716, 3.54564044550159908604074229449, 4.79398390903061347168667336836, 5.48606368860077848093138244442, 6.26999615018351715976654751606, 7.27310786296161343026635193469, 8.293548948025989544772181035186, 8.989189283626341452122201068909, 9.539258770388698648818139756227

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