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Properties

Label 2-380-20.7-c1-0-37
Degree $2$
Conductor $380$
Sign $0.804 + 0.593i$
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  + (−1.23 + 0.683i)2-s + (0.494 + 0.494i)3-s + (1.06 − 1.69i)4-s + (1.31 − 1.81i)5-s + (−0.950 − 0.274i)6-s + (−0.327 + 0.327i)7-s + (−0.165 + 2.82i)8-s − 2.51i·9-s + (−0.385 + 3.13i)10-s − 2.94i·11-s + (1.36 − 0.309i)12-s + (0.287 − 0.287i)13-s + (0.181 − 0.629i)14-s + (1.54 − 0.247i)15-s + (−1.72 − 3.60i)16-s + (−1.74 − 1.74i)17-s + ⋯
L(s)  = 1  + (−0.875 + 0.483i)2-s + (0.285 + 0.285i)3-s + (0.533 − 0.845i)4-s + (0.586 − 0.810i)5-s + (−0.387 − 0.112i)6-s + (−0.123 + 0.123i)7-s + (−0.0584 + 0.998i)8-s − 0.836i·9-s + (−0.121 + 0.992i)10-s − 0.888i·11-s + (0.393 − 0.0892i)12-s + (0.0797 − 0.0797i)13-s + (0.0486 − 0.168i)14-s + (0.398 − 0.0639i)15-s + (−0.430 − 0.902i)16-s + (−0.422 − 0.422i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.804 + 0.593i$
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: \((2,\ 380,\ (\ :1/2),\ 0.804 + 0.593i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.971871 - 0.319592i\)
\(L(\frac12)\) \(\approx\) \(0.971871 - 0.319592i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.23 - 0.683i)T \)
5 \( 1 + (-1.31 + 1.81i)T \)
19 \( 1 + T \)
good3 \( 1 + (-0.494 - 0.494i)T + 3iT^{2} \)
7 \( 1 + (0.327 - 0.327i)T - 7iT^{2} \)
11 \( 1 + 2.94iT - 11T^{2} \)
13 \( 1 + (-0.287 + 0.287i)T - 13iT^{2} \)
17 \( 1 + (1.74 + 1.74i)T + 17iT^{2} \)
23 \( 1 + (0.570 + 0.570i)T + 23iT^{2} \)
29 \( 1 + 1.85iT - 29T^{2} \)
31 \( 1 - 1.12iT - 31T^{2} \)
37 \( 1 + (-4.08 - 4.08i)T + 37iT^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 + (-7.09 - 7.09i)T + 43iT^{2} \)
47 \( 1 + (-2.57 + 2.57i)T - 47iT^{2} \)
53 \( 1 + (-5.67 + 5.67i)T - 53iT^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 - 7.69T + 61T^{2} \)
67 \( 1 + (9.51 - 9.51i)T - 67iT^{2} \)
71 \( 1 + 9.19iT - 71T^{2} \)
73 \( 1 + (4.82 - 4.82i)T - 73iT^{2} \)
79 \( 1 + 0.164T + 79T^{2} \)
83 \( 1 + (1.98 + 1.98i)T + 83iT^{2} \)
89 \( 1 - 2.27iT - 89T^{2} \)
97 \( 1 + (4.13 + 4.13i)T + 97iT^{2} \)
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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

−11.04884318906777686225525645719, −10.01171099302248532336126352063, −9.207499151416508815679307850576, −8.771643060661518646740020742416, −7.76770905269166483543699535607, −6.37365908448799579985609671182, −5.78944410080198394295519258723, −4.43849178619570342182450876418, −2.68254517170523021301270248069, −0.912744270929532088383371867880, 1.86285167920877983778765682720, 2.69374992504772740845882181189, 4.19320748040702036028544966947, 5.95248408567350137497768455766, 7.12065255058908648513554363576, 7.62824807820011470679703317338, 8.837482474326336573138609673023, 9.681647893380046523314308855574, 10.59867721644167306546589051763, 11.02284130187919145750859206424

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