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Properties

Label 2-380-380.7-c1-0-13
Degree $2$
Conductor $380$
Sign $-0.493 - 0.869i$
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  + (1.36 − 0.366i)2-s + (−0.633 + 2.36i)3-s + (1.73 − i)4-s + (−2.23 + 0.133i)5-s + 3.46i·6-s + (−3.46 + 3.46i)7-s + (1.99 − 2i)8-s + (−2.59 − 1.50i)9-s + (−2.99 + i)10-s + 1.73i·11-s + (1.26 + 4.73i)12-s + (−0.366 − 1.36i)13-s + (−3.46 + 5.99i)14-s + (1.09 − 5.36i)15-s + (1.99 − 3.46i)16-s + (−1.09 + 4.09i)17-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (−0.366 + 1.36i)3-s + (0.866 − 0.5i)4-s + (−0.998 + 0.0599i)5-s + 1.41i·6-s + (−1.30 + 1.30i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.500i)9-s + (−0.948 + 0.316i)10-s + 0.522i·11-s + (0.366 + 1.36i)12-s + (−0.101 − 0.378i)13-s + (−0.925 + 1.60i)14-s + (0.283 − 1.38i)15-s + (0.499 − 0.866i)16-s + (−0.266 + 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.493 - 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.493 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.493 - 0.869i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.493 - 0.869i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.722348 + 1.23971i\)
\(L(\frac12)\) \(\approx\) \(0.722348 + 1.23971i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
5 \( 1 + (2.23 - 0.133i)T \)
19 \( 1 + (-2.59 - 3.5i)T \)
good3 \( 1 + (0.633 - 2.36i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (3.46 - 3.46i)T - 7iT^{2} \)
11 \( 1 - 1.73iT - 11T^{2} \)
13 \( 1 + (0.366 + 1.36i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (1.09 - 4.09i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (-4.73 + 1.26i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.866 + 0.5i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.66iT - 31T^{2} \)
37 \( 1 + (-2 - 2i)T + 37iT^{2} \)
41 \( 1 + (-4 - 6.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.26 - 4.73i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-3.16 - 11.8i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.09 + 4.09i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.866 + 1.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.633 + 2.36i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.5 + 0.866i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-10.9 - 2.92i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.33 - 7.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.92 + 6.92i)T + 83iT^{2} \)
89 \( 1 + (-0.866 - 0.5i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.46 + 5.46i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.61198905145039522056014853154, −10.94482569706006345816373786686, −9.915066408730872724983291178662, −9.356438902618081873387287424305, −7.900994954921216854713538298264, −6.45790479362452439118583012005, −5.65661039334509123620342753139, −4.61227935865390320227937718770, −3.69699172784419110073451774031, −2.82000551729012040702297177174, 0.73731258284675205070941863614, 2.93642470467031870699449352504, 3.90633415054521044022320438037, 5.24270114453039340868982476978, 6.69590141912643334997691692505, 7.02253322885734794416303091653, 7.56061302029567500377161245750, 8.978265609033418181750650311209, 10.62361648806331253068759909695, 11.39893767663301502154702081392

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