↑

Properties

Label 2-1452-11.5-c1-0-16
Degree $2$
Conductor $1452$
Sign $-0.780 + 0.625i$
Analytic cond. $11.5942$
Root an. cond. $3.40503$
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  + (−0.809 − 0.587i)3-s + (0.927 − 2.85i)5-s + (1.61 − 1.17i)7-s + (0.309 + 0.951i)9-s + (−1.54 − 4.75i)13-s + (−2.42 + 1.76i)15-s + (0.927 − 2.85i)17-s + (−3.23 − 2.35i)19-s − 2·21-s + 6·23-s + (−3.23 − 2.35i)25-s + (0.309 − 0.951i)27-s + (−7.28 + 5.29i)29-s + (2.47 + 7.60i)31-s + (−1.85 − 5.70i)35-s + ⋯
L(s)  = 1  + (−0.467 − 0.339i)3-s + (0.414 − 1.27i)5-s + (0.611 − 0.444i)7-s + (0.103 + 0.317i)9-s + (−0.428 − 1.31i)13-s + (−0.626 + 0.455i)15-s + (0.224 − 0.691i)17-s + (−0.742 − 0.539i)19-s − 0.436·21-s + 1.25·23-s + (−0.647 − 0.470i)25-s + (0.0594 − 0.183i)27-s + (−1.35 + 0.982i)29-s + (0.444 + 1.36i)31-s + (−0.313 − 0.964i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $-0.780 + 0.625i$
Analytic conductor: \(11.5942\)
Root analytic conductor: \(3.40503\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1452} (1237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1452,\ (\ :1/2),\ -0.780 + 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.382336511\)
\(L(\frac12)\) \(\approx\) \(1.382336511\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 \)
good5 \( 1 + (-0.927 + 2.85i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.61 + 1.17i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.54 + 4.75i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.927 + 2.85i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.23 + 2.35i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + (7.28 - 5.29i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.47 - 7.60i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-5.66 + 4.11i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (2.42 + 1.76i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (9.70 + 7.05i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.78 + 8.55i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-4.85 + 3.52i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.32 - 13.3i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + (1.85 - 5.70i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (8.09 - 5.87i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.47 + 7.60i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (0.309 + 0.951i)T + (-78.4 + 57.0i)T^{2} \)
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

−9.078331258235101958275292977945, −8.474566886047735411521041644182, −7.55270878901987565886564468624, −6.89598998511603986259286598115, −5.54665966373925615283305268173, −5.18400760782351472082452931971, −4.42190014902423272107041287620, −2.94014646322986127770829905357, −1.54039025860521134051055403407, −0.60161462246718921934469511787, 1.78584735296495116201920185717, 2.70207520985327139372045103785, 3.96580509598774655557504116800, 4.77081389489820886998217105834, 5.99392944506426822906749641236, 6.34195704172914934441825370449, 7.33476070755550439916367737829, 8.154532653690741713822211889148, 9.339841328857801952097163498943, 9.761304012047632835673431979480

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