↑

Properties

Label 2-1911-1911.1475-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.690 + 0.722i$
Analytic cond. $0.953713$
Root an. cond. $0.976582$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.433 − 0.900i)3-s + (−0.680 + 0.733i)4-s + (0.365 − 0.930i)7-s + (−0.623 + 0.781i)9-s + (0.955 + 0.294i)12-s + (0.563 + 0.826i)13-s + (−0.0747 − 0.997i)16-s + (0.839 + 0.839i)19-s + (−0.997 + 0.0747i)21-s + (0.149 − 0.988i)25-s + (0.974 + 0.222i)27-s + (0.433 + 0.900i)28-s + (0.438 − 1.63i)31-s + (−0.149 − 0.988i)36-s + (−0.0747 − 0.00279i)37-s + ⋯
L(s)  = 1  + (−0.433 − 0.900i)3-s + (−0.680 + 0.733i)4-s + (0.365 − 0.930i)7-s + (−0.623 + 0.781i)9-s + (0.955 + 0.294i)12-s + (0.563 + 0.826i)13-s + (−0.0747 − 0.997i)16-s + (0.839 + 0.839i)19-s + (−0.997 + 0.0747i)21-s + (0.149 − 0.988i)25-s + (0.974 + 0.222i)27-s + (0.433 + 0.900i)28-s + (0.438 − 1.63i)31-s + (−0.149 − 0.988i)36-s + (−0.0747 − 0.00279i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.690 + 0.722i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.690 + 0.722i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8721123022\)
\(L(\frac12)\) \(\approx\) \(0.8721123022\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.433 + 0.900i)T \)
7 \( 1 + (-0.365 + 0.930i)T \)
13 \( 1 + (-0.563 - 0.826i)T \)
good2 \( 1 + (0.680 - 0.733i)T^{2} \)
5 \( 1 + (-0.149 + 0.988i)T^{2} \)
11 \( 1 + (0.974 + 0.222i)T^{2} \)
17 \( 1 + (-0.826 - 0.563i)T^{2} \)
19 \( 1 + (-0.839 - 0.839i)T + iT^{2} \)
23 \( 1 + (0.826 - 0.563i)T^{2} \)
29 \( 1 + (-0.826 - 0.563i)T^{2} \)
31 \( 1 + (-0.438 + 1.63i)T + (-0.866 - 0.5i)T^{2} \)
37 \( 1 + (0.0747 + 0.00279i)T + (0.997 + 0.0747i)T^{2} \)
41 \( 1 + (-0.149 + 0.988i)T^{2} \)
43 \( 1 + (-0.297 + 0.0222i)T + (0.988 - 0.149i)T^{2} \)
47 \( 1 + (0.294 + 0.955i)T^{2} \)
53 \( 1 + (-0.826 + 0.563i)T^{2} \)
59 \( 1 + (0.930 + 0.365i)T^{2} \)
61 \( 1 + (-1.86 + 0.425i)T + (0.900 - 0.433i)T^{2} \)
67 \( 1 + (-1.33 + 1.33i)T - iT^{2} \)
71 \( 1 + (0.563 + 0.826i)T^{2} \)
73 \( 1 + (0.751 - 0.554i)T + (0.294 - 0.955i)T^{2} \)
79 \( 1 + (-0.974 + 1.68i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.974 + 0.222i)T^{2} \)
89 \( 1 + (-0.294 + 0.955i)T^{2} \)
97 \( 1 + (0.416 - 1.55i)T + (-0.866 - 0.5i)T^{2} \)
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

−9.201552566763169990922267736154, −8.130074770489943226831667697886, −7.909590722171521324666853221990, −7.01595136998492107040247029346, −6.27586342554100139291835039551, −5.22909506587983046725283511619, −4.32936138603832815388505916633, −3.57856238011607477328970246312, −2.21817136943554473088640749119, −0.888170461778973404012581880183, 1.14212679329739973837562687115, 2.83056037683508911233972018449, 3.80075177337996091371216089032, 4.95050349430378493438705216677, 5.30033401760814892601339200195, 5.95445992009194942432175582273, 6.97515426841436917910860318702, 8.393502014824781646387541550207, 8.774404777306180495781075299144, 9.562826444005519330990501672016

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