Properties

Degree $2$
Conductor $588$
Sign $0.266 - 0.963i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)9-s + (3 + 5.19i)11-s − 2·13-s + (−2 + 3.46i)19-s + (3 − 5.19i)23-s + (2.5 + 4.33i)25-s − 0.999·27-s + 6·29-s + (4 + 6.92i)31-s + (−3 + 5.19i)33-s + (−1 + 1.73i)37-s + (−1 − 1.73i)39-s − 12·41-s − 4·43-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.166 + 0.288i)9-s + (0.904 + 1.56i)11-s − 0.554·13-s + (−0.458 + 0.794i)19-s + (0.625 − 1.08i)23-s + (0.5 + 0.866i)25-s − 0.192·27-s + 1.11·29-s + (0.718 + 1.24i)31-s + (−0.522 + 0.904i)33-s + (−0.164 + 0.284i)37-s + (−0.160 − 0.277i)39-s − 1.87·41-s − 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.266 - 0.963i$
Motivic weight: \(1\)
Character: $\chi_{588} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24517 + 0.947274i\)
\(L(\frac12)\) \(\approx\) \(1.24517 + 0.947274i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 12T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-6 + 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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

−10.47943702209173908529826735682, −10.16106844341492498816097585141, −9.116124051667141690885045631184, −8.443935749541386460275208542013, −7.19367553515283518122311309949, −6.55876832554089240168348208687, −5.02499277950116262149259163091, −4.40520587979508663278659154208, −3.15071750733572713514677841518, −1.77118414331097806789375930088, 0.914534402024743180882876203600, 2.58247034454246733586331312893, 3.63324610500658274442201888781, 4.94150342410443692419104271924, 6.18573546364636573254799723821, 6.80379843620544208445214810698, 7.978727741737425187783778722772, 8.702602132672200820601415364440, 9.458419761601227879363089501348, 10.59523025583029202558354503309

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