Properties

Label 2-380-380.379-c1-0-43
Degree $2$
Conductor $380$
Sign $0.683 + 0.729i$
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  + (1.39 − 0.221i)2-s − 2.04i·3-s + (1.90 − 0.619i)4-s + (0.665 + 2.13i)5-s + (−0.453 − 2.85i)6-s + 2.07·7-s + (2.51 − 1.28i)8-s − 1.17·9-s + (1.40 + 2.83i)10-s + 4.08i·11-s + (−1.26 − 3.88i)12-s − 4.52·13-s + (2.90 − 0.460i)14-s + (4.36 − 1.36i)15-s + (3.23 − 2.35i)16-s − 6.58i·17-s + ⋯
L(s)  = 1  + (0.987 − 0.156i)2-s − 1.17i·3-s + (0.950 − 0.309i)4-s + (0.297 + 0.954i)5-s + (−0.184 − 1.16i)6-s + 0.785·7-s + (0.890 − 0.454i)8-s − 0.392·9-s + (0.443 + 0.896i)10-s + 1.23i·11-s + (−0.365 − 1.12i)12-s − 1.25·13-s + (0.775 − 0.123i)14-s + (1.12 − 0.351i)15-s + (0.808 − 0.588i)16-s − 1.59i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 + 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.683 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.683 + 0.729i$
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: \((2,\ 380,\ (\ :1/2),\ 0.683 + 0.729i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.42026 - 1.04875i\)
\(L(\frac12)\) \(\approx\) \(2.42026 - 1.04875i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.221i)T \)
5 \( 1 + (-0.665 - 2.13i)T \)
19 \( 1 + (4.31 - 0.590i)T \)
good3 \( 1 + 2.04iT - 3T^{2} \)
7 \( 1 - 2.07T + 7T^{2} \)
11 \( 1 - 4.08iT - 11T^{2} \)
13 \( 1 + 4.52T + 13T^{2} \)
17 \( 1 + 6.58iT - 17T^{2} \)
23 \( 1 + 7.09T + 23T^{2} \)
29 \( 1 - 5.90iT - 29T^{2} \)
31 \( 1 - 1.69T + 31T^{2} \)
37 \( 1 - 1.81T + 37T^{2} \)
41 \( 1 + 1.10iT - 41T^{2} \)
43 \( 1 + 7.73T + 43T^{2} \)
47 \( 1 - 9.98T + 47T^{2} \)
53 \( 1 - 2.18T + 53T^{2} \)
59 \( 1 - 9.39T + 59T^{2} \)
61 \( 1 - 4.95T + 61T^{2} \)
67 \( 1 - 13.3iT - 67T^{2} \)
71 \( 1 + 4.67T + 71T^{2} \)
73 \( 1 + 6.18iT - 73T^{2} \)
79 \( 1 - 4.04T + 79T^{2} \)
83 \( 1 - 7.46T + 83T^{2} \)
89 \( 1 + 0.553iT - 89T^{2} \)
97 \( 1 + 12.4T + 97T^{2} \)
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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

−11.74519190513506721917362050313, −10.47377301367104685316128680677, −9.770352362317835469345771882595, −7.86616476600852292331955833634, −7.15191565557267125321277693919, −6.70735415778507551017322514862, −5.37858624387610118693707392943, −4.33753441497844683010545610516, −2.53328981351045027220246295423, −1.92386545280533936808911526355, 2.08165822484705220983609965364, 3.86849422883725166768225577746, 4.48360836027624414688477889259, 5.40224315860310932675060719385, 6.22286214043477290735230197081, 7.966614089533617144183464913843, 8.539305006683586120412756587322, 9.894995806925035194518229606823, 10.59038654330129441219197769904, 11.55028454140791561192811260112

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