↑

Properties

Label 2-2156-77.10-c1-0-5
Degree $2$
Conductor $2156$
Sign $0.426 - 0.904i$
Analytic cond. $17.2157$
Root an. cond. $4.14918$
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  + (−1.60 − 0.923i)3-s + (1.60 − 0.923i)5-s + (0.207 + 0.358i)9-s + (−2.23 − 2.44i)11-s − 5.84·13-s − 3.41·15-s + (−1.21 + 2.09i)17-s + (4.13 + 7.15i)19-s + (−2.12 − 3.67i)23-s + (−0.792 + 1.37i)25-s + 4.77i·27-s − 4.47i·29-s + (7.06 + 4.07i)31-s + (1.32 + 5.98i)33-s + (3.41 + 5.91i)37-s + ⋯
L(s)  = 1  + (−0.923 − 0.533i)3-s + (0.715 − 0.413i)5-s + (0.0690 + 0.119i)9-s + (−0.674 − 0.737i)11-s − 1.62·13-s − 0.881·15-s + (−0.293 + 0.508i)17-s + (0.947 + 1.64i)19-s + (−0.442 − 0.766i)23-s + (−0.158 + 0.274i)25-s + 0.919i·27-s − 0.830i·29-s + (1.26 + 0.732i)31-s + (0.230 + 1.04i)33-s + (0.561 + 0.972i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2156\)    =    \(2^{2} \cdot 7^{2} \cdot 11\)
Sign: $0.426 - 0.904i$
Analytic conductor: \(17.2157\)
Root analytic conductor: \(4.14918\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2156} (2089, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2156,\ (\ :1/2),\ 0.426 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6030837891\)
\(L(\frac12)\) \(\approx\) \(0.6030837891\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 + (2.23 + 2.44i)T \)
good3 \( 1 + (1.60 + 0.923i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.60 + 0.923i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 5.84T + 13T^{2} \)
17 \( 1 + (1.21 - 2.09i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.13 - 7.15i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.12 + 3.67i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.47iT - 29T^{2} \)
31 \( 1 + (-7.06 - 4.07i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.41 - 5.91i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 9.26T + 41T^{2} \)
43 \( 1 - 1.85iT - 43T^{2} \)
47 \( 1 + (-4.25 + 2.45i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.53 - 6.12i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.47 + 2.00i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.21 - 2.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.53 + 9.58i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.75T + 71T^{2} \)
73 \( 1 + (5.34 - 9.25i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-11.6 + 6.70i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.84T + 83T^{2} \)
89 \( 1 + (-3.86 + 2.23i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.35iT - 97T^{2} \)
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

−9.361498898301545447962337982968, −8.252398697884258963799464387671, −7.72873041158819036350919047556, −6.64010992800712577694081013164, −6.03688487822513837377258108883, −5.37871820000042513984114971986, −4.72433886217457433822222105899, −3.32614977666911909735788117357, −2.19852240790898094355514764745, −1.06409868353984020759810844362, 0.26120230318499306612644704569, 2.21470322723820414844673784289, 2.81379959274154704848980542700, 4.41473470304177018579762903901, 5.08153327803172700149031915784, 5.48524377564650213493068631194, 6.61607579635947561099650084619, 7.22121908468530676916105365202, 8.011385737100841185227400994482, 9.422287469483145431767563811540

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