Properties

Label 2-380-5.4-c1-0-9
Degree $2$
Conductor $380$
Sign $-1$
Analytic cond. $3.03431$
Root an. cond. $1.74192$
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  − 2.28i·3-s − 2.23·5-s − 2.82i·7-s − 2.23·9-s − 5.23·11-s + 4.03i·13-s + 5.11i·15-s + 1.08i·17-s + 19-s − 6.47·21-s − 7.40i·23-s + 5.00·25-s − 1.74i·27-s − 4.47·29-s − 4·31-s + ⋯
L(s)  = 1  − 1.32i·3-s − 0.999·5-s − 1.06i·7-s − 0.745·9-s − 1.57·11-s + 1.11i·13-s + 1.32i·15-s + 0.262i·17-s + 0.229·19-s − 1.41·21-s − 1.54i·23-s + 1.00·25-s − 0.336i·27-s − 0.830·29-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-1$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(-0.649728i\)
\(L(\frac12)\) \(\approx\) \(-0.649728i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + 2.23T \)
19 \( 1 - T \)
good3 \( 1 + 2.28iT - 3T^{2} \)
7 \( 1 + 2.82iT - 7T^{2} \)
11 \( 1 + 5.23T + 11T^{2} \)
13 \( 1 - 4.03iT - 13T^{2} \)
17 \( 1 - 1.08iT - 17T^{2} \)
23 \( 1 + 7.40iT - 23T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 6.86iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 8.48iT - 43T^{2} \)
47 \( 1 + 8.48iT - 47T^{2} \)
53 \( 1 + 6.86iT - 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 - 1.70T + 61T^{2} \)
67 \( 1 + 1.62iT - 67T^{2} \)
71 \( 1 + 1.52T + 71T^{2} \)
73 \( 1 + 13.7iT - 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 + 5.24iT - 83T^{2} \)
89 \( 1 - 14.9T + 89T^{2} \)
97 \( 1 + 4.44iT - 97T^{2} \)
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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

−11.02145493194901859885902023156, −10.23970892972287980582582391544, −8.692576938098605556743543492367, −7.78090352218386426196057524651, −7.29162301662007195457792502083, −6.48724781096419677549843793191, −4.90856598343253198185912743718, −3.73970299622832026259108721631, −2.15682926332740490322782216036, −0.41877617072356171753293268452, 2.83598098964990282632676514655, 3.71737393637441797905767167907, 5.16370653955465194142215282014, 5.45324970390636644061466481841, 7.40744180709567331132945366982, 8.191259016259086839951194102281, 9.133209419868067713714478655336, 10.06705981094389970561365768546, 10.82859514215923855997715543253, 11.60450118938691524018470651589

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