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Properties

Label 2-1380-115.22-c1-0-14
Degree $2$
Conductor $1380$
Sign $0.129 + 0.991i$
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 + (−2.09 + 0.768i)5-s + (−0.0117 + 0.0117i)7-s − 1.00i·9-s + 1.72i·11-s + (−2.37 + 2.37i)13-s + (0.941 − 2.02i)15-s + (1.66 − 1.66i)17-s − 6.73·19-s − 0.0165i·21-s + (−0.0541 − 4.79i)23-s + (3.81 − 3.22i)25-s + (0.707 + 0.707i)27-s − 0.894i·29-s + 6.90·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.939 + 0.343i)5-s + (−0.00443 + 0.00443i)7-s − 0.333i·9-s + 0.520i·11-s + (−0.657 + 0.657i)13-s + (0.243 − 0.523i)15-s + (0.404 − 0.404i)17-s − 1.54·19-s − 0.00361i·21-s + (−0.0112 − 0.999i)23-s + (0.763 − 0.645i)25-s + (0.136 + 0.136i)27-s − 0.166i·29-s + 1.23·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.129 + 0.991i$
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: \((2,\ 1380,\ (\ :1/2),\ 0.129 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4813333736\)
\(L(\frac12)\) \(\approx\) \(0.4813333736\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.09 - 0.768i)T \)
23 \( 1 + (0.0541 + 4.79i)T \)
good7 \( 1 + (0.0117 - 0.0117i)T - 7iT^{2} \)
11 \( 1 - 1.72iT - 11T^{2} \)
13 \( 1 + (2.37 - 2.37i)T - 13iT^{2} \)
17 \( 1 + (-1.66 + 1.66i)T - 17iT^{2} \)
19 \( 1 + 6.73T + 19T^{2} \)
29 \( 1 + 0.894iT - 29T^{2} \)
31 \( 1 - 6.90T + 31T^{2} \)
37 \( 1 + (0.0575 - 0.0575i)T - 37iT^{2} \)
41 \( 1 - 0.278T + 41T^{2} \)
43 \( 1 + (-4.31 - 4.31i)T + 43iT^{2} \)
47 \( 1 + (7.61 + 7.61i)T + 47iT^{2} \)
53 \( 1 + (9.64 + 9.64i)T + 53iT^{2} \)
59 \( 1 + 15.1iT - 59T^{2} \)
61 \( 1 + 5.91iT - 61T^{2} \)
67 \( 1 + (1.99 - 1.99i)T - 67iT^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 + (-7.55 + 7.55i)T - 73iT^{2} \)
79 \( 1 - 3.91T + 79T^{2} \)
83 \( 1 + (12.2 + 12.2i)T + 83iT^{2} \)
89 \( 1 + 6.61T + 89T^{2} \)
97 \( 1 + (-9.80 + 9.80i)T - 97iT^{2} \)
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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

−9.566307905753201643068059661264, −8.491828226020649994753037663433, −7.84145257362852156328670422740, −6.77276534246747839898335906058, −6.35958282302503131184248755150, −4.79967919507124330572830754156, −4.49712834437715595989820868638, −3.38217917980610702315956763662, −2.20386527233660480708669406182, −0.23619884499033771232223467599, 1.12181482796380178133052999031, 2.67223910221279363495035617533, 3.80545740844616365075343738075, 4.71832119755127993248528144553, 5.63416073818155848557107594299, 6.49784870869176348248731590425, 7.46373749349583511356474649507, 8.082331222550155275954333593070, 8.727069762698494814466271950871, 9.830673386550159895497018578346

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