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Properties

Label 2-380-380.327-c1-0-46
Degree $2$
Conductor $380$
Sign $0.0386 + 0.999i$
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  + (1.09 − 0.895i)2-s + (0.235 − 0.336i)3-s + (0.397 − 1.96i)4-s + (2.06 − 0.853i)5-s + (−0.0431 − 0.578i)6-s + (−0.371 − 1.38i)7-s + (−1.31 − 2.50i)8-s + (0.968 + 2.66i)9-s + (1.49 − 2.78i)10-s + (−1.39 + 0.806i)11-s + (−0.565 − 0.594i)12-s + (−1.24 − 1.77i)13-s + (−1.64 − 1.18i)14-s + (0.199 − 0.895i)15-s + (−3.68 − 1.55i)16-s + (1.48 + 3.18i)17-s + ⋯
L(s)  = 1  + (0.774 − 0.632i)2-s + (0.135 − 0.194i)3-s + (0.198 − 0.980i)4-s + (0.924 − 0.381i)5-s + (−0.0176 − 0.236i)6-s + (−0.140 − 0.524i)7-s + (−0.466 − 0.884i)8-s + (0.322 + 0.886i)9-s + (0.474 − 0.880i)10-s + (−0.421 + 0.243i)11-s + (−0.163 − 0.171i)12-s + (−0.344 − 0.492i)13-s + (−0.440 − 0.316i)14-s + (0.0515 − 0.231i)15-s + (−0.920 − 0.389i)16-s + (0.360 + 0.772i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0386 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0386 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.0386 + 0.999i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.0386 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68161 - 1.61775i\)
\(L(\frac12)\) \(\approx\) \(1.68161 - 1.61775i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.09 + 0.895i)T \)
5 \( 1 + (-2.06 + 0.853i)T \)
19 \( 1 + (-0.135 - 4.35i)T \)
good3 \( 1 + (-0.235 + 0.336i)T + (-1.02 - 2.81i)T^{2} \)
7 \( 1 + (0.371 + 1.38i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.39 - 0.806i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.24 + 1.77i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (-1.48 - 3.18i)T + (-10.9 + 13.0i)T^{2} \)
23 \( 1 + (0.283 + 3.24i)T + (-22.6 + 3.99i)T^{2} \)
29 \( 1 + (-0.834 - 2.29i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (5.96 + 3.44i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.32 - 2.32i)T + 37iT^{2} \)
41 \( 1 + (-0.909 - 5.16i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-1.51 - 0.132i)T + (42.3 + 7.46i)T^{2} \)
47 \( 1 + (0.338 - 0.726i)T + (-30.2 - 36.0i)T^{2} \)
53 \( 1 + (1.43 - 0.125i)T + (52.1 - 9.20i)T^{2} \)
59 \( 1 + (-0.719 - 0.261i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.724 - 0.607i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (6.15 - 13.1i)T + (-43.0 - 51.3i)T^{2} \)
71 \( 1 + (-1.97 - 2.34i)T + (-12.3 + 69.9i)T^{2} \)
73 \( 1 + (-0.596 - 0.417i)T + (24.9 + 68.5i)T^{2} \)
79 \( 1 + (-1.90 - 10.7i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-12.3 + 3.30i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (17.2 + 3.05i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (0.691 + 1.48i)T + (-62.3 + 74.3i)T^{2} \)
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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

−10.99400698946253638113977047551, −10.24723337394560531259090140469, −9.800007829724219920650435758583, −8.370079668445993176178991077389, −7.24027754741953079600312187273, −6.00663582333631464323139277419, −5.19666873530526990745059891275, −4.13776753430425463297999941480, −2.62693706449245656941716693367, −1.49305960344125437280820965864, 2.42685592539307652911435596838, 3.46680850857341189924093228287, 4.90439350449063363469175659427, 5.78578061616344517423984173106, 6.70210753723359495873885593888, 7.50897391024799862472973249947, 9.043350402630604370491124808337, 9.409226865550096808392927624438, 10.75768535814665600083151965592, 11.80714669096233131928182862146

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