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Properties

Label 2-1380-115.68-c1-0-15
Degree $2$
Conductor $1380$
Sign $-0.357 + 0.933i$
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 + (−1.35 − 1.77i)5-s + (0.243 + 0.243i)7-s + 1.00i·9-s − 5.94i·11-s + (4.09 + 4.09i)13-s + (−0.298 + 2.21i)15-s + (3.99 + 3.99i)17-s + 6.79·19-s − 0.343i·21-s + (−1.51 − 4.54i)23-s + (−1.32 + 4.82i)25-s + (0.707 − 0.707i)27-s − 6.73i·29-s − 4.13·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.606 − 0.795i)5-s + (0.0918 + 0.0918i)7-s + 0.333i·9-s − 1.79i·11-s + (1.13 + 1.13i)13-s + (−0.0771 + 0.572i)15-s + (0.967 + 0.967i)17-s + 1.55·19-s − 0.0750i·21-s + (−0.316 − 0.948i)23-s + (−0.264 + 0.964i)25-s + (0.136 − 0.136i)27-s − 1.25i·29-s − 0.743·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.357 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.357 + 0.933i$
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: \((2,\ 1380,\ (\ :1/2),\ -0.357 + 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.215750127\)
\(L(\frac12)\) \(\approx\) \(1.215750127\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.35 + 1.77i)T \)
23 \( 1 + (1.51 + 4.54i)T \)
good7 \( 1 + (-0.243 - 0.243i)T + 7iT^{2} \)
11 \( 1 + 5.94iT - 11T^{2} \)
13 \( 1 + (-4.09 - 4.09i)T + 13iT^{2} \)
17 \( 1 + (-3.99 - 3.99i)T + 17iT^{2} \)
19 \( 1 - 6.79T + 19T^{2} \)
29 \( 1 + 6.73iT - 29T^{2} \)
31 \( 1 + 4.13T + 31T^{2} \)
37 \( 1 + (7.79 + 7.79i)T + 37iT^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + (-2.50 + 2.50i)T - 43iT^{2} \)
47 \( 1 + (-4.12 + 4.12i)T - 47iT^{2} \)
53 \( 1 + (-2.71 + 2.71i)T - 53iT^{2} \)
59 \( 1 - 12.6iT - 59T^{2} \)
61 \( 1 - 1.15iT - 61T^{2} \)
67 \( 1 + (1.96 + 1.96i)T + 67iT^{2} \)
71 \( 1 + 1.14T + 71T^{2} \)
73 \( 1 + (-3.22 - 3.22i)T + 73iT^{2} \)
79 \( 1 + 2.34T + 79T^{2} \)
83 \( 1 + (-7.74 + 7.74i)T - 83iT^{2} \)
89 \( 1 + 7.32T + 89T^{2} \)
97 \( 1 + (1.43 + 1.43i)T + 97iT^{2} \)
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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

−8.976441528637533348984751669005, −8.569144684702578154059873759006, −7.85803282518026402444042088838, −6.86758590526711946771129977465, −5.79883965985299475224510927581, −5.45563877674464925404295680652, −4.02270781713292971393285103061, −3.44327203261504305942812365236, −1.65130372895639696867112623246, −0.60001220526105598721831741119, 1.34881034598964614940621224012, 3.11142463660814436355693084065, 3.61415699855110688638199388746, 4.92166879791713033524532090053, 5.48006722327007960591328096445, 6.72122071971612742531978687165, 7.40921454137140919966619381104, 7.950180311935143463882482388169, 9.250753982430959056444103349392, 9.985810888445696726532719203060

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