↑

Properties

Label 2-476-7.5-c2-0-13
Degree $2$
Conductor $476$
Sign $0.485 + 0.874i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−3.34 + 1.93i)3-s + (6.37 + 3.68i)5-s + (−6.40 − 2.81i)7-s + (2.95 − 5.12i)9-s + (−5.25 − 9.10i)11-s − 16.4i·13-s − 28.4·15-s + (−3.57 + 2.06i)17-s + (0.965 + 0.557i)19-s + (26.8 − 2.96i)21-s + (−8.54 + 14.8i)23-s + (14.5 + 25.2i)25-s − 11.9i·27-s + 2.59·29-s + (16.5 − 9.56i)31-s + ⋯
L(s)  = 1  + (−1.11 + 0.643i)3-s + (1.27 + 0.736i)5-s + (−0.915 − 0.401i)7-s + (0.328 − 0.568i)9-s + (−0.477 − 0.827i)11-s − 1.26i·13-s − 1.89·15-s + (−0.210 + 0.121i)17-s + (0.0508 + 0.0293i)19-s + (1.27 − 0.141i)21-s + (−0.371 + 0.643i)23-s + (0.583 + 1.01i)25-s − 0.441i·27-s + 0.0893·29-s + (0.534 − 0.308i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $0.485 + 0.874i$
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: \((2,\ 476,\ (\ :1),\ 0.485 + 0.874i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8223466975\)
\(L(\frac12)\) \(\approx\) \(0.8223466975\)
\(L(2)\) 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 \( 1 + (6.40 + 2.81i)T \)
17 \( 1 + (3.57 - 2.06i)T \)
good3 \( 1 + (3.34 - 1.93i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-6.37 - 3.68i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (5.25 + 9.10i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 16.4iT - 169T^{2} \)
19 \( 1 + (-0.965 - 0.557i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (8.54 - 14.8i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 2.59T + 841T^{2} \)
31 \( 1 + (-16.5 + 9.56i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-28.0 + 48.5i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 71.7iT - 1.68e3T^{2} \)
43 \( 1 - 8.00T + 1.84e3T^{2} \)
47 \( 1 + (37.8 + 21.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (1.39 + 2.42i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (11.0 - 6.35i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-24.3 - 14.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-4.02 - 6.96i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 57.1T + 5.04e3T^{2} \)
73 \( 1 + (-43.5 + 25.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-63.7 + 110. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 141. iT - 6.88e3T^{2} \)
89 \( 1 + (11.9 + 6.90i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 94.8iT - 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.50698769454544256115512217961, −10.12915523100362170471234461276, −9.250719537231275721730114826172, −7.78750172362550386815623683226, −6.53313887380775971574762647915, −5.84538788580355279768975278422, −5.33386111288796088421177664203, −3.71677701459304261931569662484, −2.56835515822173175733856541832, −0.40122568369607460988620421141, 1.31239826391526390008148963159, 2.49555753862265315810292379406, 4.58330552747951526049017168075, 5.42485841977942057384794121848, 6.46109568775603804430356204584, 6.67723452485261620797410268220, 8.292280157532220572920738790426, 9.569283173868656183453791511812, 9.740379305306070484796786473929, 11.06477998883021024610850082448

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