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Properties

Label 2-380-20.7-c1-0-44
Degree $2$
Conductor $380$
Sign $-0.935 - 0.354i$
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  + (−1.07 − 0.923i)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 + (1.51 − 2.38i)8-s − 0.360i·9-s + (−3.09 − 0.640i)10-s − 4.11i·11-s + (1.93 − 2.60i)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  + (−0.757 − 0.653i)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 + (0.535 − 0.844i)8-s − 0.120i·9-s + (−0.979 − 0.202i)10-s − 1.24i·11-s + (0.558 − 0.753i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.935 - 0.354i$
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: \((2,\ 380,\ (\ :1/2),\ -0.935 - 0.354i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0618549 + 0.337672i\)
\(L(\frac12)\) \(\approx\) \(0.0618549 + 0.337672i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.07 + 0.923i)T \)
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

−10.89831902210709278736271351864, −9.790449411167419464359583687251, −9.084093606380590623425070168918, −8.484041403619286200926364192973, −6.74345701257362474569897005407, −6.37312749098068662176791545072, −5.09732968089439933654243739443, −3.19103802786610890987201137158, −2.01031910586848278180950961790, −0.29474581718659850757500816290, 2.11526137857516629953589349449, 4.16601195759201380370620409117, 5.30467016962870010469805397894, 6.33379007918573836158255928979, 6.97610558030851891341800348509, 8.039366423559056921290606977990, 9.754357969452773552212298998523, 9.819022461161734408371528222137, 10.51675298303352138573741099497, 11.36732585479727890293509738240

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