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Properties

Label 2-476-7.5-c2-0-9
Degree $2$
Conductor $476$
Sign $0.0633 - 0.997i$
Analytic cond. $12.9700$
Root an. cond. $3.60139$
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  + (0.557 − 0.321i)3-s + (5.89 + 3.40i)5-s + (3.87 + 5.82i)7-s + (−4.29 + 7.43i)9-s + (−1.20 − 2.08i)11-s + 16.8i·13-s + 4.37·15-s + (−3.57 + 2.06i)17-s + (−27.5 − 15.9i)19-s + (4.03 + 1.99i)21-s + (−6.26 + 10.8i)23-s + (10.6 + 18.4i)25-s + 11.3i·27-s − 32.2·29-s + (41.3 − 23.8i)31-s + ⋯
L(s)  = 1  + (0.185 − 0.107i)3-s + (1.17 + 0.680i)5-s + (0.553 + 0.832i)7-s + (−0.477 + 0.826i)9-s + (−0.109 − 0.189i)11-s + 1.29i·13-s + 0.291·15-s + (−0.210 + 0.121i)17-s + (−1.45 − 0.837i)19-s + (0.192 + 0.0952i)21-s + (−0.272 + 0.471i)23-s + (0.425 + 0.737i)25-s + 0.418i·27-s − 1.11·29-s + (1.33 − 0.770i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $0.0633 - 0.997i$
Analytic conductor: \(12.9700\)
Root analytic conductor: \(3.60139\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{476} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 476,\ (\ :1),\ 0.0633 - 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.071755392\)
\(L(\frac12)\) \(\approx\) \(2.071755392\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-3.87 - 5.82i)T \)
17 \( 1 + (3.57 - 2.06i)T \)
good3 \( 1 + (-0.557 + 0.321i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-5.89 - 3.40i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (1.20 + 2.08i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 16.8iT - 169T^{2} \)
19 \( 1 + (27.5 + 15.9i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (6.26 - 10.8i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 32.2T + 841T^{2} \)
31 \( 1 + (-41.3 + 23.8i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-22.9 + 39.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 21.1iT - 1.68e3T^{2} \)
43 \( 1 + 48.7T + 1.84e3T^{2} \)
47 \( 1 + (-71.4 - 41.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-35.9 - 62.2i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-89.5 + 51.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-78.8 - 45.5i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-38.0 - 65.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 18.7T + 5.04e3T^{2} \)
73 \( 1 + (51.0 - 29.4i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-65.0 + 112. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 36.2iT - 6.88e3T^{2} \)
89 \( 1 + (111. + 64.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 6.25iT - 9.40e3T^{2} \)
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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

−11.09511308679132396994422312365, −10.13397621418941058575069437866, −9.113097115639279987769089566438, −8.516570143684553165377890478560, −7.29114302744498455844534584153, −6.22977811748562405588603382156, −5.54689353081207497238030177070, −4.32182687079770249095839412275, −2.40213619833406502703752921424, −2.11840419542259672754487416529, 0.807685911014261909543721503817, 2.20114717909862105534332476741, 3.70880114070806528305018422385, 4.89881629520590434210854259105, 5.82968392019951616903633266636, 6.73911974188444908001906760883, 8.188442467572809496909288974586, 8.631617578164735418939216602105, 9.976880698519854731628924336785, 10.20080993587283746903026764931

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