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Properties

Label 2-380-380.379-c1-0-53
Degree $2$
Conductor $380$
Sign $-0.958 + 0.284i$
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  + (0.846 − 1.13i)2-s − 3.11i·3-s + (−0.568 − 1.91i)4-s + 2.23·5-s + (−3.52 − 2.63i)6-s + (−2.65 − 0.978i)8-s − 6.67·9-s + (1.89 − 2.53i)10-s + 2.92i·11-s + (−5.96 + 1.76i)12-s + 6.79·13-s − 6.95i·15-s + (−3.35 + 2.17i)16-s + (−5.65 + 7.56i)18-s + 4.35i·19-s + (−1.27 − 4.28i)20-s + ⋯
L(s)  = 1  + (0.598 − 0.801i)2-s − 1.79i·3-s + (−0.284 − 0.958i)4-s + 0.999·5-s + (−1.43 − 1.07i)6-s + (−0.938 − 0.345i)8-s − 2.22·9-s + (0.598 − 0.801i)10-s + 0.883i·11-s + (−1.72 + 0.510i)12-s + 1.88·13-s − 1.79i·15-s + (−0.838 + 0.544i)16-s + (−1.33 + 1.78i)18-s + 0.999i·19-s + (−0.284 − 0.958i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.284i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.958 + 0.284i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.958 + 0.284i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.289856 - 1.99816i\)
\(L(\frac12)\) \(\approx\) \(0.289856 - 1.99816i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.846 + 1.13i)T \)
5 \( 1 - 2.23T \)
19 \( 1 - 4.35iT \)
good3 \( 1 + 3.11iT - 3T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 2.92iT - 11T^{2} \)
13 \( 1 - 6.79T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 12.1T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 6.04T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 15.5T + 61T^{2} \)
67 \( 1 + 16.0iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 2.99T + 97T^{2} \)
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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

−11.17144122400775140613454882235, −10.28123565065907540848260172787, −9.099872427013777726025593520814, −8.181440802096175654670373295100, −6.75250059897538427638159703518, −6.19183840543160516661853515005, −5.28811000967257858853921633402, −3.44948376432868299248315705012, −2.02316253771729066816255870837, −1.36269486951045195331853012536, 3.05662483446825610801703526235, 3.88258900138381113188432228176, 5.07279773084316696285326919312, 5.74418656654470966990536659793, 6.60028474595845226527685774651, 8.602132355630940267552017091850, 8.758704189461801576684767264890, 9.852321691958548756803078508861, 10.87862381702048164650341903786, 11.43652106071440726657056709329

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