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Properties

Label 2-1452-11.9-c1-0-1
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

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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: $\chi_{1452} (493, \cdot )$
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 \( 1 + (0.309 - 0.951i)T + (-78.4 - 57.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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