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Properties

Label 2-1380-115.22-c1-0-7
Degree $2$
Conductor $1380$
Sign $0.268 - 0.963i$
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 + (−0.0653 − 2.23i)5-s + (−0.522 + 0.522i)7-s − 1.00i·9-s + 3.66i·11-s + (0.950 − 0.950i)13-s + (1.62 + 1.53i)15-s + (−3.43 + 3.43i)17-s − 1.65·19-s − 0.738i·21-s + (4.78 + 0.329i)23-s + (−4.99 + 0.292i)25-s + (0.707 + 0.707i)27-s − 1.81i·29-s − 1.24·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.0292 − 0.999i)5-s + (−0.197 + 0.197i)7-s − 0.333i·9-s + 1.10i·11-s + (0.263 − 0.263i)13-s + (0.420 + 0.396i)15-s + (−0.832 + 0.832i)17-s − 0.379·19-s − 0.161i·21-s + (0.997 + 0.0687i)23-s + (−0.998 + 0.0584i)25-s + (0.136 + 0.136i)27-s − 0.336i·29-s − 0.223·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.268 - 0.963i$
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: \((2,\ 1380,\ (\ :1/2),\ 0.268 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.092336113\)
\(L(\frac12)\) \(\approx\) \(1.092336113\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0653 + 2.23i)T \)
23 \( 1 + (-4.78 - 0.329i)T \)
good7 \( 1 + (0.522 - 0.522i)T - 7iT^{2} \)
11 \( 1 - 3.66iT - 11T^{2} \)
13 \( 1 + (-0.950 + 0.950i)T - 13iT^{2} \)
17 \( 1 + (3.43 - 3.43i)T - 17iT^{2} \)
19 \( 1 + 1.65T + 19T^{2} \)
29 \( 1 + 1.81iT - 29T^{2} \)
31 \( 1 + 1.24T + 31T^{2} \)
37 \( 1 + (2.81 - 2.81i)T - 37iT^{2} \)
41 \( 1 - 12.2T + 41T^{2} \)
43 \( 1 + (-7.63 - 7.63i)T + 43iT^{2} \)
47 \( 1 + (-6.98 - 6.98i)T + 47iT^{2} \)
53 \( 1 + (1.34 + 1.34i)T + 53iT^{2} \)
59 \( 1 - 1.65iT - 59T^{2} \)
61 \( 1 + 2.65iT - 61T^{2} \)
67 \( 1 + (5.90 - 5.90i)T - 67iT^{2} \)
71 \( 1 - 6.37T + 71T^{2} \)
73 \( 1 + (11.2 - 11.2i)T - 73iT^{2} \)
79 \( 1 - 2.17T + 79T^{2} \)
83 \( 1 + (-8.40 - 8.40i)T + 83iT^{2} \)
89 \( 1 + 8.81T + 89T^{2} \)
97 \( 1 + (0.249 - 0.249i)T - 97iT^{2} \)
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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

−9.517990823608065368871377442095, −9.164013322164995192059889751170, −8.243321440265202753097365046967, −7.34760124327394351692528976669, −6.28419199883581547193219651337, −5.56671273298019214810138093908, −4.53299152616438839318598398640, −4.12109889223213118245403162196, −2.56385919910502511962432148898, −1.20879065789751900080092617154, 0.52167756422734225042733383480, 2.21937569313455515965169397644, 3.18624274112684703318043867767, 4.19450171923236464489014064788, 5.46273998824193244967548016197, 6.21165120546570280488224547431, 6.98075843151843827525304292372, 7.50475380411293041717382406336, 8.695243087288374720163292439540, 9.299971908543148372389956112118

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