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Properties

Label 2-381-127.20-c2-0-15
Degree $2$
Conductor $381$
Sign $0.699 - 0.714i$
Analytic cond. $10.3814$
Root an. cond. $3.22203$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.80·2-s + (−1.5 + 0.866i)3-s − 0.731·4-s + 3.62i·5-s + (2.71 − 1.56i)6-s + (6.52 − 3.76i)7-s + 8.55·8-s + (1.5 − 2.59i)9-s − 6.54i·10-s + (−7.69 + 13.3i)11-s + (1.09 − 0.633i)12-s + (−7.51 − 13.0i)13-s + (−11.7 + 6.80i)14-s + (−3.13 − 5.43i)15-s − 12.5·16-s + (16.3 − 28.2i)17-s + ⋯
L(s)  = 1  − 0.903·2-s + (−0.5 + 0.288i)3-s − 0.182·4-s + 0.724i·5-s + (0.451 − 0.260i)6-s + (0.931 − 0.537i)7-s + 1.06·8-s + (0.166 − 0.288i)9-s − 0.654i·10-s + (−0.699 + 1.21i)11-s + (0.0914 − 0.0527i)12-s + (−0.577 − 1.00i)13-s + (−0.842 + 0.486i)14-s + (−0.209 − 0.362i)15-s − 0.783·16-s + (0.960 − 1.66i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(381\)    =    \(3 \cdot 127\)
Sign: $0.699 - 0.714i$
Analytic conductor: \(10.3814\)
Root analytic conductor: \(3.22203\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{381} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 381,\ (\ :1),\ 0.699 - 0.714i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.717821 + 0.301801i\)
\(L(\frac12)\) \(\approx\) \(0.717821 + 0.301801i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 0.866i)T \)
127 \( 1 + (-126. - 5.97i)T \)
good2 \( 1 + 1.80T + 4T^{2} \)
5 \( 1 - 3.62iT - 25T^{2} \)
7 \( 1 + (-6.52 + 3.76i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (7.69 - 13.3i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (7.51 + 13.0i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-16.3 + 28.2i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + 17.8T + 361T^{2} \)
23 \( 1 + (-8.37 + 4.83i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-38.8 - 22.4i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-21.1 - 36.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (3.18 - 5.51i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (18.5 - 32.0i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-17.9 - 10.3i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 - 58.5T + 2.20e3T^{2} \)
53 \( 1 + (-1.58 - 0.913i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-16.4 - 9.47i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + 4.15T + 3.72e3T^{2} \)
67 \( 1 + (9.77 - 5.64i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-57.7 - 100. i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + 50.6T + 5.32e3T^{2} \)
79 \( 1 + (66.1 + 114. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-138. - 80.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 92.9iT - 7.92e3T^{2} \)
97 \( 1 + (-16.1 + 9.34i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.67229735821564661824395902337, −10.45062196787083699358581687883, −9.695271503626252936926292328773, −8.432925242809223937179194189001, −7.51021224788600639080850864413, −6.96007973621028067826635261123, −5.04974220942373090738700465292, −4.68295554115902937882724957147, −2.77992566723287797502584126552, −0.939337450566762699342040991596, 0.72859356138645563468287854098, 2.00223403811983099743274140459, 4.25915713954053000155156827188, 5.17909184899668247583103617840, 6.17244521930247322692505751195, 7.70180476986696409656237240576, 8.393212841771989242937540406699, 8.872712997769305912864068730270, 10.16727487964075311640251210486, 10.85520511253733646977073913904

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