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Properties

Label 2-380-20.7-c1-0-5
Degree $2$
Conductor $380$
Sign $-0.856 + 0.515i$
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  + (−0.283 + 1.38i)2-s + (1.20 + 1.20i)3-s + (−1.83 − 0.786i)4-s + (−2.06 + 0.854i)5-s + (−2.00 + 1.32i)6-s + (−1.52 + 1.52i)7-s + (1.61 − 2.32i)8-s − 0.119i·9-s + (−0.597 − 3.10i)10-s + 4.01i·11-s + (−1.26 − 3.15i)12-s + (−1.50 + 1.50i)13-s + (−1.67 − 2.53i)14-s + (−3.50 − 1.45i)15-s + (2.76 + 2.89i)16-s + (−3.94 − 3.94i)17-s + ⋯
L(s)  = 1  + (−0.200 + 0.979i)2-s + (0.692 + 0.692i)3-s + (−0.919 − 0.393i)4-s + (−0.924 + 0.382i)5-s + (−0.817 + 0.539i)6-s + (−0.574 + 0.574i)7-s + (0.569 − 0.821i)8-s − 0.0397i·9-s + (−0.189 − 0.981i)10-s + 1.21i·11-s + (−0.364 − 0.909i)12-s + (−0.416 + 0.416i)13-s + (−0.447 − 0.678i)14-s + (−0.905 − 0.375i)15-s + (0.690 + 0.722i)16-s + (−0.957 − 0.957i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.856 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.856 + 0.515i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.856 + 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.192069 - 0.691574i\)
\(L(\frac12)\) \(\approx\) \(0.192069 - 0.691574i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.283 - 1.38i)T \)
5 \( 1 + (2.06 - 0.854i)T \)
19 \( 1 + T \)
good3 \( 1 + (-1.20 - 1.20i)T + 3iT^{2} \)
7 \( 1 + (1.52 - 1.52i)T - 7iT^{2} \)
11 \( 1 - 4.01iT - 11T^{2} \)
13 \( 1 + (1.50 - 1.50i)T - 13iT^{2} \)
17 \( 1 + (3.94 + 3.94i)T + 17iT^{2} \)
23 \( 1 + (4.78 + 4.78i)T + 23iT^{2} \)
29 \( 1 - 7.97iT - 29T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
37 \( 1 + (-6.01 - 6.01i)T + 37iT^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 + (1.96 + 1.96i)T + 43iT^{2} \)
47 \( 1 + (-2.21 + 2.21i)T - 47iT^{2} \)
53 \( 1 + (5.27 - 5.27i)T - 53iT^{2} \)
59 \( 1 + 5.37T + 59T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 + (0.631 - 0.631i)T - 67iT^{2} \)
71 \( 1 - 6.42iT - 71T^{2} \)
73 \( 1 + (2.26 - 2.26i)T - 73iT^{2} \)
79 \( 1 + 4.54T + 79T^{2} \)
83 \( 1 + (-4.42 - 4.42i)T + 83iT^{2} \)
89 \( 1 - 8.81iT - 89T^{2} \)
97 \( 1 + (-8.06 - 8.06i)T + 97iT^{2} \)
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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

−12.09548693783966918726308010715, −10.62229814972605258131681831657, −9.738958178434621322658606766948, −9.043556496732536298962402569581, −8.309487739795155862339566218815, −7.08268872525417468947136000616, −6.57103693767722705118072489867, −4.83795933529252635312038291072, −4.16302578316188881666643114496, −2.82099331807761372190002996688, 0.46555354333530545270373096728, 2.23038949885717051353195882539, 3.51012423058662661723721476165, 4.30072231274142584052021390078, 5.98558368940303623408080172305, 7.66251607699483198299244662924, 7.989538095897079126434357094279, 8.906636588105066629019103737957, 9.927632776083530744698250779335, 11.03613582018518417963208482434

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