Properties

Label 2-380-20.7-c1-0-26
Degree $2$
Conductor $380$
Sign $0.792 + 0.610i$
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  + (−0.923 − 1.07i)2-s + (1.14 + 1.14i)3-s + (−0.292 + 1.97i)4-s + (1.95 − 1.08i)5-s + (0.168 − 2.29i)6-s + (2.70 − 2.70i)7-s + (2.38 − 1.51i)8-s − 0.360i·9-s + (−2.96 − 1.08i)10-s + 4.11i·11-s + (−2.60 + 1.93i)12-s + (−1.95 + 1.95i)13-s + (−5.40 − 0.397i)14-s + (3.49 + 0.995i)15-s + (−3.82 − 1.15i)16-s + (−4.15 − 4.15i)17-s + ⋯
L(s)  = 1  + (−0.653 − 0.757i)2-s + (0.663 + 0.663i)3-s + (−0.146 + 0.989i)4-s + (0.873 − 0.486i)5-s + (0.0688 − 0.935i)6-s + (1.02 − 1.02i)7-s + (0.844 − 0.535i)8-s − 0.120i·9-s + (−0.939 − 0.343i)10-s + 1.24i·11-s + (−0.753 + 0.558i)12-s + (−0.541 + 0.541i)13-s + (−1.44 − 0.106i)14-s + (0.902 + 0.256i)15-s + (−0.957 − 0.289i)16-s + (−1.00 − 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.792 + 0.610i$
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: \((2,\ 380,\ (\ :1/2),\ 0.792 + 0.610i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39639 - 0.475443i\)
\(L(\frac12)\) \(\approx\) \(1.39639 - 0.475443i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 + 1.07i)T \)
5 \( 1 + (-1.95 + 1.08i)T \)
19 \( 1 - T \)
good3 \( 1 + (-1.14 - 1.14i)T + 3iT^{2} \)
7 \( 1 + (-2.70 + 2.70i)T - 7iT^{2} \)
11 \( 1 - 4.11iT - 11T^{2} \)
13 \( 1 + (1.95 - 1.95i)T - 13iT^{2} \)
17 \( 1 + (4.15 + 4.15i)T + 17iT^{2} \)
23 \( 1 + (-3.01 - 3.01i)T + 23iT^{2} \)
29 \( 1 - 2.35iT - 29T^{2} \)
31 \( 1 + 5.48iT - 31T^{2} \)
37 \( 1 + (-3.69 - 3.69i)T + 37iT^{2} \)
41 \( 1 + 9.82T + 41T^{2} \)
43 \( 1 + (-8.96 - 8.96i)T + 43iT^{2} \)
47 \( 1 + (-0.482 + 0.482i)T - 47iT^{2} \)
53 \( 1 + (2.71 - 2.71i)T - 53iT^{2} \)
59 \( 1 + 9.02T + 59T^{2} \)
61 \( 1 + 4.23T + 61T^{2} \)
67 \( 1 + (6.49 - 6.49i)T - 67iT^{2} \)
71 \( 1 - 14.9iT - 71T^{2} \)
73 \( 1 + (-7.29 + 7.29i)T - 73iT^{2} \)
79 \( 1 + 3.64T + 79T^{2} \)
83 \( 1 + (8.32 + 8.32i)T + 83iT^{2} \)
89 \( 1 + 9.76iT - 89T^{2} \)
97 \( 1 + (-0.208 - 0.208i)T + 97iT^{2} \)
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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

−11.08874681858771333766943505184, −10.05707992352304513646487159269, −9.519251353712092594962103844885, −8.922878440286360697092908875532, −7.73642446600126476854567584682, −6.90490781797565196550002058421, −4.70650667633820014532275938884, −4.37031531763799008296696023902, −2.68768123738960650118024849891, −1.45259975309468995770666735374, 1.70904411722886761020645585283, 2.66113184712958998364615145846, 5.04452731221334498728441461849, 5.84898359009542780911595117104, 6.81040509616047307633611818493, 7.929137752684906104767932105917, 8.571397058742484058792684307372, 9.165495235757134924789227082547, 10.61633322752963822045960139409, 11.00437597437717272922179551148

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