↑

Properties

Label 2-476-119.108-c1-0-6
Degree $2$
Conductor $476$
Sign $0.102 + 0.994i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.58 − 0.879i)3-s + (0.0752 − 1.14i)5-s + (2.14 + 1.55i)7-s + (3.55 + 2.72i)9-s + (3.21 + 2.81i)11-s + (−4.37 − 4.37i)13-s + (−1.20 + 2.90i)15-s + (2.49 − 3.28i)17-s + (0.0295 − 0.00389i)19-s + (−4.17 − 5.90i)21-s + (−1.61 − 4.75i)23-s + (3.64 + 0.480i)25-s + (−2.24 − 3.36i)27-s + (−4.74 − 3.16i)29-s + (2.56 − 7.56i)31-s + ⋯
L(s)  = 1  + (−1.49 − 0.507i)3-s + (0.0336 − 0.513i)5-s + (0.809 + 0.587i)7-s + (1.18 + 0.908i)9-s + (0.968 + 0.849i)11-s + (−1.21 − 1.21i)13-s + (−0.310 + 0.750i)15-s + (0.605 − 0.796i)17-s + (0.00677 − 0.000892i)19-s + (−0.911 − 1.28i)21-s + (−0.336 − 0.991i)23-s + (0.729 + 0.0960i)25-s + (−0.432 − 0.646i)27-s + (−0.880 − 0.588i)29-s + (0.461 − 1.35i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $0.102 + 0.994i$
Analytic conductor: \(3.80087\)
Root analytic conductor: \(1.94958\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{476} (465, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 476,\ (\ :1/2),\ 0.102 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.633083 - 0.571403i\)
\(L(\frac12)\) \(\approx\) \(0.633083 - 0.571403i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.14 - 1.55i)T \)
17 \( 1 + (-2.49 + 3.28i)T \)
good3 \( 1 + (2.58 + 0.879i)T + (2.38 + 1.82i)T^{2} \)
5 \( 1 + (-0.0752 + 1.14i)T + (-4.95 - 0.652i)T^{2} \)
11 \( 1 + (-3.21 - 2.81i)T + (1.43 + 10.9i)T^{2} \)
13 \( 1 + (4.37 + 4.37i)T + 13iT^{2} \)
19 \( 1 + (-0.0295 + 0.00389i)T + (18.3 - 4.91i)T^{2} \)
23 \( 1 + (1.61 + 4.75i)T + (-18.2 + 14.0i)T^{2} \)
29 \( 1 + (4.74 + 3.16i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 + (-2.56 + 7.56i)T + (-24.5 - 18.8i)T^{2} \)
37 \( 1 + (6.86 + 7.82i)T + (-4.82 + 36.6i)T^{2} \)
41 \( 1 + (-1.42 + 0.955i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (-6.87 + 2.84i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + (1.56 - 5.85i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.17 + 2.43i)T + (13.7 - 51.1i)T^{2} \)
59 \( 1 + (1.15 - 8.80i)T + (-56.9 - 15.2i)T^{2} \)
61 \( 1 + (-2.90 + 5.88i)T + (-37.1 - 48.3i)T^{2} \)
67 \( 1 + (-0.368 - 0.212i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.59 + 0.515i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-13.2 + 6.53i)T + (44.4 - 57.9i)T^{2} \)
79 \( 1 + (10.5 - 3.58i)T + (62.6 - 48.0i)T^{2} \)
83 \( 1 + (9.53 + 3.94i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-1.43 - 0.383i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.667 - 0.998i)T + (-37.1 - 89.6i)T^{2} \)
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

−11.01906104506034143159580861526, −10.03874035846960379485307446398, −9.109537265110413342560261111918, −7.78969957190400478320129736151, −7.12507240275115667419572018590, −5.89110130809905332369899713375, −5.22917750280823242871051031190, −4.45007824558336499907788669247, −2.22512903457984704137177408828, −0.70572620393786240936224182625, 1.40763016090903178985447129538, 3.62991456488170611251516831890, 4.63906705317277244745430906295, 5.49432651514044414375141652487, 6.57844155053894288460150182978, 7.19521287513816479400146489668, 8.595864341194992143640622178990, 9.779030397675823948614187486403, 10.49458501829745071531841270148, 11.29579472927625017809684368071

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