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Properties

Label 2-380-76.27-c1-0-24
Degree $2$
Conductor $380$
Sign $0.988 - 0.152i$
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  + (1.33 − 0.460i)2-s + (−0.435 + 0.755i)3-s + (1.57 − 1.23i)4-s + (0.5 − 0.866i)5-s + (−0.235 + 1.21i)6-s + 4.89i·7-s + (1.54 − 2.37i)8-s + (1.11 + 1.93i)9-s + (0.270 − 1.38i)10-s + 0.00879i·11-s + (0.241 + 1.72i)12-s + (0.420 − 0.242i)13-s + (2.25 + 6.55i)14-s + (0.435 + 0.755i)15-s + (0.972 − 3.87i)16-s + (3.74 − 6.49i)17-s + ⋯
L(s)  = 1  + (0.945 − 0.325i)2-s + (−0.251 + 0.435i)3-s + (0.788 − 0.615i)4-s + (0.223 − 0.387i)5-s + (−0.0962 + 0.494i)6-s + 1.85i·7-s + (0.545 − 0.838i)8-s + (0.373 + 0.646i)9-s + (0.0854 − 0.438i)10-s + 0.00265i·11-s + (0.0697 + 0.498i)12-s + (0.116 − 0.0673i)13-s + (0.602 + 1.75i)14-s + (0.112 + 0.194i)15-s + (0.243 − 0.969i)16-s + (0.908 − 1.57i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.988 - 0.152i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.988 - 0.152i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31730 + 0.178228i\)
\(L(\frac12)\) \(\approx\) \(2.31730 + 0.178228i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 + 0.460i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (4.34 + 0.362i)T \)
good3 \( 1 + (0.435 - 0.755i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 4.89iT - 7T^{2} \)
11 \( 1 - 0.00879iT - 11T^{2} \)
13 \( 1 + (-0.420 + 0.242i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.74 + 6.49i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-5.37 + 3.10i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.57 - 3.79i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.94T + 31T^{2} \)
37 \( 1 - 2.82iT - 37T^{2} \)
41 \( 1 + (-1.66 - 0.962i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.58 + 1.48i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.00 + 0.579i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.12 - 1.22i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.18 - 8.97i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.851 + 1.47i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.12 + 5.40i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.62 + 11.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.76 + 8.25i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.52 + 2.64i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.01iT - 83T^{2} \)
89 \( 1 + (4.11 - 2.37i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.51 + 3.76i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.49063780196538618701490539082, −10.72606842991128145303434235512, −9.604662584392561530017015646548, −8.895234133727660744144428651022, −7.47427488417507131826516051706, −6.15080445137027709482792597796, −5.24616918054736868810613927647, −4.79568910289461013989949435939, −3.10115042481251040718942703774, −1.99539229336463021452658514639, 1.54000239454751298545748005781, 3.58309884235493687102737113602, 4.11390192125687390039684140099, 5.68314575313353347871132305971, 6.62015179091717693181676322147, 7.24036774646235575913887615432, 8.066611162741405834759683119753, 9.753615593461507728615440518107, 10.75170682485917926437950261519, 11.26040180650573091015754641361

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