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Properties

Label 2-476-119.100-c1-0-10
Degree $2$
Conductor $476$
Sign $-0.0362 + 0.999i$
Analytic cond. $3.80087$
Root an. cond. $1.94958$
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  + (1.75 − 2.28i)3-s + (−0.0892 + 0.678i)5-s + (1.39 − 2.25i)7-s + (−1.37 − 5.12i)9-s + (−2.70 + 0.355i)11-s − 1.91i·13-s + (1.39 + 1.39i)15-s + (3.97 − 1.09i)17-s + (−7.44 + 1.99i)19-s + (−2.70 − 7.12i)21-s + (3.96 + 5.16i)23-s + (4.37 + 1.17i)25-s + (−6.14 − 2.54i)27-s + (8.44 − 3.49i)29-s + (0.277 − 0.361i)31-s + ⋯
L(s)  = 1  + (1.01 − 1.32i)3-s + (−0.0399 + 0.303i)5-s + (0.525 − 0.850i)7-s + (−0.457 − 1.70i)9-s + (−0.814 + 0.107i)11-s − 0.530i·13-s + (0.359 + 0.359i)15-s + (0.964 − 0.264i)17-s + (−1.70 + 0.457i)19-s + (−0.590 − 1.55i)21-s + (0.826 + 1.07i)23-s + (0.875 + 0.234i)25-s + (−1.18 − 0.489i)27-s + (1.56 − 0.649i)29-s + (0.0497 − 0.0648i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0362 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0362 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $-0.0362 + 0.999i$
Analytic conductor: \(3.80087\)
Root analytic conductor: \(1.94958\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{476} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 476,\ (\ :1/2),\ -0.0362 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33768 - 1.38706i\)
\(L(\frac12)\) \(\approx\) \(1.33768 - 1.38706i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-1.39 + 2.25i)T \)
17 \( 1 + (-3.97 + 1.09i)T \)
good3 \( 1 + (-1.75 + 2.28i)T + (-0.776 - 2.89i)T^{2} \)
5 \( 1 + (0.0892 - 0.678i)T + (-4.82 - 1.29i)T^{2} \)
11 \( 1 + (2.70 - 0.355i)T + (10.6 - 2.84i)T^{2} \)
13 \( 1 + 1.91iT - 13T^{2} \)
19 \( 1 + (7.44 - 1.99i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.96 - 5.16i)T + (-5.95 + 22.2i)T^{2} \)
29 \( 1 + (-8.44 + 3.49i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-0.277 + 0.361i)T + (-8.02 - 29.9i)T^{2} \)
37 \( 1 + (7.40 + 0.974i)T + (35.7 + 9.57i)T^{2} \)
41 \( 1 + (-3.96 - 1.64i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (5.63 - 5.63i)T - 43iT^{2} \)
47 \( 1 + (7.20 + 4.15i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.206 - 0.769i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.80 - 1.55i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.639 - 0.490i)T + (15.7 - 58.9i)T^{2} \)
67 \( 1 + (-7.59 - 13.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.336 - 0.811i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-2.84 - 2.18i)T + (18.8 + 70.5i)T^{2} \)
79 \( 1 + (-2.21 - 2.88i)T + (-20.4 + 76.3i)T^{2} \)
83 \( 1 + (-3.81 - 3.81i)T + 83iT^{2} \)
89 \( 1 + (-9.12 - 5.26i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.24 + 2.17i)T + (68.5 - 68.5i)T^{2} \)
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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

−10.71018981877957179046010833340, −9.954093823157116398638066756471, −8.516361613301924509827136461629, −8.039278842208090957160829417899, −7.26889529840309537712966186976, −6.51702470781619755607528185970, −5.06394139542098445800976217164, −3.53123077851181125683282498546, −2.51108447985337794125602846700, −1.17017680419381091252401643237, 2.28215674561093583323396586153, 3.25989524362416185735288446560, 4.65554135615827542975984710606, 5.04678280743605357011367784431, 6.58636925678063558965064983740, 8.251319626985917913175580854956, 8.521871735493402208642980654536, 9.266978257663712932945933365799, 10.44763950557269514811651282234, 10.76416261321629581108232290328

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