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Properties

Label 2-380-20.7-c1-0-1
Degree $2$
Conductor $380$
Sign $-0.994 - 0.103i$
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  + (0.647 + 1.25i)2-s + (−1.78 − 1.78i)3-s + (−1.16 + 1.62i)4-s + (0.0829 − 2.23i)5-s + (1.09 − 3.40i)6-s + (−0.904 + 0.904i)7-s + (−2.79 − 0.408i)8-s + 3.39i·9-s + (2.86 − 1.34i)10-s + 4.92i·11-s + (4.98 − 0.831i)12-s + (−4.64 + 4.64i)13-s + (−1.72 − 0.551i)14-s + (−4.14 + 3.84i)15-s + (−1.29 − 3.78i)16-s + (−3.50 − 3.50i)17-s + ⋯
L(s)  = 1  + (0.457 + 0.889i)2-s + (−1.03 − 1.03i)3-s + (−0.581 + 0.813i)4-s + (0.0370 − 0.999i)5-s + (0.445 − 1.39i)6-s + (−0.341 + 0.341i)7-s + (−0.989 − 0.144i)8-s + 1.13i·9-s + (0.905 − 0.424i)10-s + 1.48i·11-s + (1.44 − 0.240i)12-s + (−1.28 + 1.28i)13-s + (−0.460 − 0.147i)14-s + (−1.07 + 0.993i)15-s + (−0.324 − 0.945i)16-s + (−0.850 − 0.850i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.994 - 0.103i$
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: \((2,\ 380,\ (\ :1/2),\ -0.994 - 0.103i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0120912 + 0.232823i\)
\(L(\frac12)\) \(\approx\) \(0.0120912 + 0.232823i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.647 - 1.25i)T \)
5 \( 1 + (-0.0829 + 2.23i)T \)
19 \( 1 + T \)
good3 \( 1 + (1.78 + 1.78i)T + 3iT^{2} \)
7 \( 1 + (0.904 - 0.904i)T - 7iT^{2} \)
11 \( 1 - 4.92iT - 11T^{2} \)
13 \( 1 + (4.64 - 4.64i)T - 13iT^{2} \)
17 \( 1 + (3.50 + 3.50i)T + 17iT^{2} \)
23 \( 1 + (-2.65 - 2.65i)T + 23iT^{2} \)
29 \( 1 - 4.23iT - 29T^{2} \)
31 \( 1 + 6.52iT - 31T^{2} \)
37 \( 1 + (4.68 + 4.68i)T + 37iT^{2} \)
41 \( 1 - 3.99T + 41T^{2} \)
43 \( 1 + (1.41 + 1.41i)T + 43iT^{2} \)
47 \( 1 + (0.801 - 0.801i)T - 47iT^{2} \)
53 \( 1 + (2.72 - 2.72i)T - 53iT^{2} \)
59 \( 1 + 13.9T + 59T^{2} \)
61 \( 1 - 6.61T + 61T^{2} \)
67 \( 1 + (1.41 - 1.41i)T - 67iT^{2} \)
71 \( 1 - 0.666iT - 71T^{2} \)
73 \( 1 + (-9.09 + 9.09i)T - 73iT^{2} \)
79 \( 1 + 7.09T + 79T^{2} \)
83 \( 1 + (6.54 + 6.54i)T + 83iT^{2} \)
89 \( 1 + 6.08iT - 89T^{2} \)
97 \( 1 + (-1.52 - 1.52i)T + 97iT^{2} \)
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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

−12.19989959214451357590774093596, −11.44308037774028027588714446002, −9.528230738380507966166632667996, −9.087335460447535695853013560585, −7.47003019463606413880379289395, −7.11677782827995092049320310804, −6.13338695814091434752541318628, −5.03585259224905465616295249954, −4.48352538067230257780798785573, −2.08389051310091181830622862208, 0.14347159738999094511125820077, 2.81391656853781015160626588884, 3.72476134417638037783581788009, 4.90716624739772350683907882431, 5.81531607437407539466880185140, 6.62899551793009804755513310055, 8.361279652760057484084503241364, 9.709408966409797219559872694605, 10.43085946073292535752361092623, 10.78567584031038890976023261956

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