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Properties

Label 2-380-380.379-c1-0-47
Degree $2$
Conductor $380$
Sign $-0.976 + 0.216i$
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  + (0.440 − 1.34i)2-s − 0.562i·3-s + (−1.61 − 1.18i)4-s + (1.40 − 1.73i)5-s + (−0.756 − 0.247i)6-s − 3.12·7-s + (−2.29 + 1.64i)8-s + 2.68·9-s + (−1.71 − 2.65i)10-s − 3.83i·11-s + (−0.665 + 0.907i)12-s − 2.34·13-s + (−1.37 + 4.20i)14-s + (−0.977 − 0.792i)15-s + (1.20 + 3.81i)16-s − 0.296i·17-s + ⋯
L(s)  = 1  + (0.311 − 0.950i)2-s − 0.324i·3-s + (−0.806 − 0.591i)4-s + (0.629 − 0.777i)5-s + (−0.308 − 0.101i)6-s − 1.18·7-s + (−0.812 + 0.582i)8-s + 0.894·9-s + (−0.542 − 0.839i)10-s − 1.15i·11-s + (−0.192 + 0.262i)12-s − 0.650·13-s + (−0.367 + 1.12i)14-s + (−0.252 − 0.204i)15-s + (0.300 + 0.953i)16-s − 0.0718i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.216i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.976 + 0.216i$
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: \((2,\ 380,\ (\ :1/2),\ -0.976 + 0.216i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.142675 - 1.29960i\)
\(L(\frac12)\) \(\approx\) \(0.142675 - 1.29960i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.440 + 1.34i)T \)
5 \( 1 + (-1.40 + 1.73i)T \)
19 \( 1 + (4.35 + 0.167i)T \)
good3 \( 1 + 0.562iT - 3T^{2} \)
7 \( 1 + 3.12T + 7T^{2} \)
11 \( 1 + 3.83iT - 11T^{2} \)
13 \( 1 + 2.34T + 13T^{2} \)
17 \( 1 + 0.296iT - 17T^{2} \)
23 \( 1 - 5.14T + 23T^{2} \)
29 \( 1 - 6.81iT - 29T^{2} \)
31 \( 1 - 4.76T + 31T^{2} \)
37 \( 1 + 1.81T + 37T^{2} \)
41 \( 1 + 12.0iT - 41T^{2} \)
43 \( 1 + 0.171T + 43T^{2} \)
47 \( 1 - 6.96T + 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 + 7.33T + 59T^{2} \)
61 \( 1 - 3.27T + 61T^{2} \)
67 \( 1 - 9.46iT - 67T^{2} \)
71 \( 1 - 9.45T + 71T^{2} \)
73 \( 1 + 8.23iT - 73T^{2} \)
79 \( 1 - 7.49T + 79T^{2} \)
83 \( 1 - 6.37T + 83T^{2} \)
89 \( 1 - 3.49iT - 89T^{2} \)
97 \( 1 - 11.9T + 97T^{2} \)
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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

−10.78989690807128505661314583481, −10.13686876078052844469456018150, −9.213902314066418726614681359514, −8.606538859601431044796623414711, −6.92356756438331776534415263049, −5.93394982984402984086471898103, −4.90469026717348661608308452150, −3.66753767144191668138406316759, −2.37725101420519221589533336713, −0.813761611189285569888383201119, 2.63062918258658386831657907531, 3.97438211498732365773900375337, 4.97893543232582048186949771554, 6.36818073938307860705154863947, 6.80357218037357262379819583179, 7.71974891596430977432030089189, 9.288477221765575170601417533745, 9.779653576172947575667619146106, 10.46408591450435447589878700872, 12.09244118826239991455211438996

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