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Properties

Label 2-1152-72.59-c1-0-20
Degree $2$
Conductor $1152$
Sign $0.904 + 0.426i$
Analytic cond. $9.19876$
Root an. cond. $3.03294$
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  + (−1.65 − 0.524i)3-s + (1.57 − 2.72i)5-s + (−2.21 + 1.27i)7-s + (2.44 + 1.73i)9-s + (−2.02 + 1.16i)11-s + (2.59 + 1.5i)13-s + (−4.02 + 3.67i)15-s + 4.24i·17-s + 8.08·19-s + (4.33 − 0.949i)21-s + (0.642 − 1.11i)23-s + (−2.44 − 4.24i)25-s + (−3.13 − 4.14i)27-s + (1.18 + 2.05i)29-s + (7.64 + 4.41i)31-s + ⋯
L(s)  = 1  + (−0.953 − 0.302i)3-s + (0.703 − 1.21i)5-s + (−0.837 + 0.483i)7-s + (0.816 + 0.577i)9-s + (−0.609 + 0.351i)11-s + (0.720 + 0.416i)13-s + (−1.03 + 0.948i)15-s + 1.02i·17-s + 1.85·19-s + (0.944 − 0.207i)21-s + (0.133 − 0.232i)23-s + (−0.489 − 0.848i)25-s + (−0.603 − 0.797i)27-s + (0.219 + 0.380i)29-s + (1.37 + 0.792i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.904 + 0.426i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.904 + 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.248524756\)
\(L(\frac12)\) \(\approx\) \(1.248524756\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.65 + 0.524i)T \)
good5 \( 1 + (-1.57 + 2.72i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.21 - 1.27i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.02 - 1.16i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.59 - 1.5i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.24iT - 17T^{2} \)
19 \( 1 - 8.08T + 19T^{2} \)
23 \( 1 + (-0.642 + 1.11i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.18 - 2.05i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.64 - 4.41i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.34iT - 37T^{2} \)
41 \( 1 + (8.17 + 4.71i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.11 + 1.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.78 + 8.29i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.34T + 53T^{2} \)
59 \( 1 + (-1.11 - 0.642i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.79 + 4.5i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.204 - 0.353i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.5T + 71T^{2} \)
73 \( 1 + 1.55T + 73T^{2} \)
79 \( 1 + (-8.93 + 5.15i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.93 - 1.69i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 6.14iT - 89T^{2} \)
97 \( 1 + (-6.62 - 11.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.853468710701004517873875849481, −8.968211249379452176152873442293, −8.224921379769340460099515559401, −7.02715679277237912725530381465, −6.25413576333142108599025014839, −5.40181623332359104401543521498, −4.99992800673341256351911122882, −3.62674309343804300249433637556, −2.03303952702598207497855417831, −0.929242422156112675213655549884, 0.889608418954882258603150983521, 2.85829928872823006442092288375, 3.44441221616050849180766219726, 4.89936263453615376606426625650, 5.75372329575563399931200679600, 6.48773287451121238458917144526, 7.04124526309446789657753620643, 8.040334754883267680104041410861, 9.643434054133858398155761660337, 9.874148577101749137851558487125

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