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Properties

Label 2-476-7.5-c2-0-7
Degree $2$
Conductor $476$
Sign $0.915 + 0.402i$
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  + (−2.25 + 1.30i)3-s + (−5.00 − 2.89i)5-s + (−6.80 + 1.63i)7-s + (−1.10 + 1.91i)9-s + (1.75 + 3.04i)11-s + 7.32i·13-s + 15.0·15-s + (−3.57 + 2.06i)17-s + (−9.14 − 5.27i)19-s + (13.2 − 12.5i)21-s + (17.5 − 30.4i)23-s + (4.23 + 7.32i)25-s − 29.2i·27-s + 38.7·29-s + (21.1 − 12.1i)31-s + ⋯
L(s)  = 1  + (−0.751 + 0.434i)3-s + (−1.00 − 0.578i)5-s + (−0.972 + 0.233i)7-s + (−0.123 + 0.213i)9-s + (0.159 + 0.276i)11-s + 0.563i·13-s + 1.00·15-s + (−0.210 + 0.121i)17-s + (−0.481 − 0.277i)19-s + (0.629 − 0.597i)21-s + (0.764 − 1.32i)23-s + (0.169 + 0.293i)25-s − 1.08i·27-s + 1.33·29-s + (0.680 − 0.393i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $0.915 + 0.402i$
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: \((2,\ 476,\ (\ :1),\ 0.915 + 0.402i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6553411747\)
\(L(\frac12)\) \(\approx\) \(0.6553411747\)
\(L(2)\) 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 \( 1 + (6.80 - 1.63i)T \)
17 \( 1 + (3.57 - 2.06i)T \)
good3 \( 1 + (2.25 - 1.30i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (5.00 + 2.89i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-1.75 - 3.04i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 7.32iT - 169T^{2} \)
19 \( 1 + (9.14 + 5.27i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-17.5 + 30.4i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 38.7T + 841T^{2} \)
31 \( 1 + (-21.1 + 12.1i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-3.14 + 5.45i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 36.9iT - 1.68e3T^{2} \)
43 \( 1 - 5.49T + 1.84e3T^{2} \)
47 \( 1 + (11.7 + 6.76i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-28.2 - 48.9i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (19.1 - 11.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-14.9 - 8.61i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (0.959 + 1.66i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 39.5T + 5.04e3T^{2} \)
73 \( 1 + (-21.2 + 12.2i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-19.2 + 33.3i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 2.63iT - 6.88e3T^{2} \)
89 \( 1 + (46.5 + 26.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 162. iT - 9.40e3T^{2} \)
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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

−10.78791495374018162030775489191, −9.968588423386630834860038413357, −8.899374895072519440292808150026, −8.179635717658655548878180550732, −6.86546604439833850458495271090, −6.09994298369625335944308439126, −4.75094621649906999158469931500, −4.23678885302999193989289615749, −2.70826218386605058680750881187, −0.46976452745895232361090741534, 0.76343834241340316502939941510, 3.02536770760399071046777973093, 3.81194897287479682705631990698, 5.33298157505154632417356911320, 6.43576329889860023790316265431, 6.96031848321764963432895726638, 7.950788372167005819987925970263, 9.060451763136673578209019419655, 10.17530995727058277623859258056, 11.00766255598407824694042607078

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