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Properties

Label 2-380-20.7-c1-0-33
Degree $2$
Conductor $380$
Sign $0.919 + 0.392i$
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  + (0.435 − 1.34i)2-s + (1.82 + 1.82i)3-s + (−1.61 − 1.17i)4-s + (2.11 + 0.724i)5-s + (3.24 − 1.65i)6-s + (1.36 − 1.36i)7-s + (−2.28 + 1.66i)8-s + 3.63i·9-s + (1.89 − 2.53i)10-s − 1.55i·11-s + (−0.814 − 5.08i)12-s + (−2.11 + 2.11i)13-s + (−1.23 − 2.42i)14-s + (2.53 + 5.17i)15-s + (1.24 + 3.80i)16-s + (0.503 + 0.503i)17-s + ⋯
L(s)  = 1  + (0.308 − 0.951i)2-s + (1.05 + 1.05i)3-s + (−0.809 − 0.586i)4-s + (0.946 + 0.324i)5-s + (1.32 − 0.676i)6-s + (0.514 − 0.514i)7-s + (−0.807 + 0.589i)8-s + 1.21i·9-s + (0.599 − 0.800i)10-s − 0.468i·11-s + (−0.235 − 1.46i)12-s + (−0.585 + 0.585i)13-s + (−0.330 − 0.647i)14-s + (0.654 + 1.33i)15-s + (0.312 + 0.950i)16-s + (0.122 + 0.122i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.392i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 + 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.919 + 0.392i$
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: \((2,\ 380,\ (\ :1/2),\ 0.919 + 0.392i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.24332 - 0.458342i\)
\(L(\frac12)\) \(\approx\) \(2.24332 - 0.458342i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.435 + 1.34i)T \)
5 \( 1 + (-2.11 - 0.724i)T \)
19 \( 1 + T \)
good3 \( 1 + (-1.82 - 1.82i)T + 3iT^{2} \)
7 \( 1 + (-1.36 + 1.36i)T - 7iT^{2} \)
11 \( 1 + 1.55iT - 11T^{2} \)
13 \( 1 + (2.11 - 2.11i)T - 13iT^{2} \)
17 \( 1 + (-0.503 - 0.503i)T + 17iT^{2} \)
23 \( 1 + (3.79 + 3.79i)T + 23iT^{2} \)
29 \( 1 - 2.83iT - 29T^{2} \)
31 \( 1 + 4.40iT - 31T^{2} \)
37 \( 1 + (2.89 + 2.89i)T + 37iT^{2} \)
41 \( 1 - 5.57T + 41T^{2} \)
43 \( 1 + (5.10 + 5.10i)T + 43iT^{2} \)
47 \( 1 + (9.15 - 9.15i)T - 47iT^{2} \)
53 \( 1 + (-5.52 + 5.52i)T - 53iT^{2} \)
59 \( 1 + 9.43T + 59T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 + (-0.237 + 0.237i)T - 67iT^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 + (7.73 - 7.73i)T - 73iT^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 + (-0.972 - 0.972i)T + 83iT^{2} \)
89 \( 1 + 4.55iT - 89T^{2} \)
97 \( 1 + (-13.9 - 13.9i)T + 97iT^{2} \)
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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

−10.88876115979900582964861037496, −10.43463311107730823179686221450, −9.557160040708421640902039248196, −9.012840596706477974576897677080, −7.945876525938160932571240912943, −6.26293096461449175933353190630, −4.95756983963743185607208265019, −4.11376497258317857205686274647, −3.01199187020389937597246880665, −1.94569425675469061523274002289, 1.80065765792047427973923339211, 3.02744857575078838765290298271, 4.77619892336637500831728475524, 5.74751001694586744538035655027, 6.78769264819453405861366489346, 7.72932882620101996941903081355, 8.381513382277843248497547154606, 9.210576288206049209243578022153, 10.07301373267672335047950836441, 11.98461766563622150363995075363

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