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Properties

Label 2-380-76.75-c1-0-28
Degree $2$
Conductor $380$
Sign $0.454 + 0.890i$
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  + (−1.40 + 0.195i)2-s + 1.89·3-s + (1.92 − 0.547i)4-s − 5-s + (−2.65 + 0.370i)6-s − 2.89i·7-s + (−2.58 + 1.14i)8-s + 0.596·9-s + (1.40 − 0.195i)10-s − 5.39i·11-s + (3.64 − 1.03i)12-s − 1.52i·13-s + (0.564 + 4.04i)14-s − 1.89·15-s + (3.40 − 2.10i)16-s + 2.35·17-s + ⋯
L(s)  = 1  + (−0.990 + 0.138i)2-s + 1.09·3-s + (0.961 − 0.273i)4-s − 0.447·5-s + (−1.08 + 0.151i)6-s − 1.09i·7-s + (−0.914 + 0.403i)8-s + 0.198·9-s + (0.442 − 0.0617i)10-s − 1.62i·11-s + (1.05 − 0.299i)12-s − 0.422i·13-s + (0.150 + 1.08i)14-s − 0.489·15-s + (0.850 − 0.526i)16-s + 0.572·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.454 + 0.890i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.454 + 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.921651 - 0.564560i\)
\(L(\frac12)\) \(\approx\) \(0.921651 - 0.564560i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.195i)T \)
5 \( 1 + T \)
19 \( 1 + (0.841 + 4.27i)T \)
good3 \( 1 - 1.89T + 3T^{2} \)
7 \( 1 + 2.89iT - 7T^{2} \)
11 \( 1 + 5.39iT - 11T^{2} \)
13 \( 1 + 1.52iT - 13T^{2} \)
17 \( 1 - 2.35T + 17T^{2} \)
23 \( 1 - 5.72iT - 23T^{2} \)
29 \( 1 - 7.31iT - 29T^{2} \)
31 \( 1 - 9.03T + 31T^{2} \)
37 \( 1 - 0.169iT - 37T^{2} \)
41 \( 1 + 10.2iT - 41T^{2} \)
43 \( 1 + 0.321iT - 43T^{2} \)
47 \( 1 - 8.93iT - 47T^{2} \)
53 \( 1 + 4.58iT - 53T^{2} \)
59 \( 1 - 6.24T + 59T^{2} \)
61 \( 1 - 14.3T + 61T^{2} \)
67 \( 1 + 2.38T + 67T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 + 2.57T + 73T^{2} \)
79 \( 1 + 4.66T + 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 + 2.58iT - 89T^{2} \)
97 \( 1 - 14.9iT - 97T^{2} \)
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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

−10.98664921455748418991887028666, −10.20625920742613535865899441759, −9.096002089889097454350844211416, −8.428707876911879536945058586233, −7.73747393136386798070043689114, −6.89898271952395737450975615594, −5.55445813221901835471946627682, −3.64833250979529975407686518411, −2.87700072444327414259948626325, −0.897471266532150468238690480183, 2.01251864266858959428596718592, 2.83307056097451013300107881570, 4.29468085238390861570314062445, 6.03451101179127578387348808874, 7.19220887506005133931374300667, 8.181157826351321409185709470647, 8.574131545529928022969330158264, 9.686085681172905245618173753737, 10.11460074886962753758113628932, 11.72576067493666147124877080370

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