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Properties

Label 2-1380-115.22-c1-0-11
Degree $2$
Conductor $1380$
Sign $0.735 - 0.677i$
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 + (1.86 + 1.22i)5-s + (−0.329 + 0.329i)7-s − 1.00i·9-s + 2.84i·11-s + (−3.80 + 3.80i)13-s + (2.18 − 0.453i)15-s + (3.63 − 3.63i)17-s + 3.13·19-s + 0.465i·21-s + (−0.0646 + 4.79i)23-s + (1.98 + 4.58i)25-s + (−0.707 − 0.707i)27-s + 2.87i·29-s + 4.47·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.835 + 0.548i)5-s + (−0.124 + 0.124i)7-s − 0.333i·9-s + 0.857i·11-s + (−1.05 + 1.05i)13-s + (0.565 − 0.117i)15-s + (0.882 − 0.882i)17-s + 0.718·19-s + 0.101i·21-s + (−0.0134 + 0.999i)23-s + (0.397 + 0.917i)25-s + (−0.136 − 0.136i)27-s + 0.534i·29-s + 0.804·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.735 - 0.677i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ 0.735 - 0.677i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.123455135\)
\(L(\frac12)\) \(\approx\) \(2.123455135\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.86 - 1.22i)T \)
23 \( 1 + (0.0646 - 4.79i)T \)
good7 \( 1 + (0.329 - 0.329i)T - 7iT^{2} \)
11 \( 1 - 2.84iT - 11T^{2} \)
13 \( 1 + (3.80 - 3.80i)T - 13iT^{2} \)
17 \( 1 + (-3.63 + 3.63i)T - 17iT^{2} \)
19 \( 1 - 3.13T + 19T^{2} \)
29 \( 1 - 2.87iT - 29T^{2} \)
31 \( 1 - 4.47T + 31T^{2} \)
37 \( 1 + (2.51 - 2.51i)T - 37iT^{2} \)
41 \( 1 - 1.80T + 41T^{2} \)
43 \( 1 + (1.58 + 1.58i)T + 43iT^{2} \)
47 \( 1 + (-1.81 - 1.81i)T + 47iT^{2} \)
53 \( 1 + (-3.43 - 3.43i)T + 53iT^{2} \)
59 \( 1 + 13.1iT - 59T^{2} \)
61 \( 1 + 1.84iT - 61T^{2} \)
67 \( 1 + (-10.9 + 10.9i)T - 67iT^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + (6.01 - 6.01i)T - 73iT^{2} \)
79 \( 1 - 16.4T + 79T^{2} \)
83 \( 1 + (5.51 + 5.51i)T + 83iT^{2} \)
89 \( 1 - 2.89T + 89T^{2} \)
97 \( 1 + (7.16 - 7.16i)T - 97iT^{2} \)
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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

−9.557592016664966706405283106658, −9.226842035873703413288198302479, −7.83633521667249946698367585036, −7.18556714662984037437172145209, −6.64023461785105446312135593041, −5.51660422302842808357659850302, −4.72897856131309109405724265356, −3.32451691570503196941965504879, −2.46632932735918855129465806263, −1.51265302484456179055451155275, 0.866245093748400328235590988345, 2.38722720286796136368863137694, 3.26327551665747227778793654875, 4.41091055827926778138520315631, 5.42407097089631339385887383763, 5.88742867505570189859868173763, 7.11272700930230765862736856275, 8.169465236928415068049783288339, 8.578600483270075617922554342828, 9.656147597410979693703972305036

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