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Properties

Label 2-459-153.106-c1-0-11
Degree $2$
Conductor $459$
Sign $0.996 + 0.0856i$
Analytic cond. $3.66513$
Root an. cond. $1.91445$
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  + (0.793 + 0.458i)2-s + (−0.580 − 1.00i)4-s + (3.41 + 0.915i)5-s + (−0.340 + 0.0913i)7-s − 2.89i·8-s + (2.29 + 2.29i)10-s + (3.14 − 0.843i)11-s + (−2.19 − 3.79i)13-s + (−0.312 − 0.0836i)14-s + (0.165 − 0.286i)16-s + (−0.593 + 4.08i)17-s + 3.53i·19-s + (−1.06 − 3.96i)20-s + (2.88 + 0.772i)22-s + (1.54 − 5.78i)23-s + ⋯
L(s)  = 1  + (0.560 + 0.323i)2-s + (−0.290 − 0.502i)4-s + (1.52 + 0.409i)5-s + (−0.128 + 0.0345i)7-s − 1.02i·8-s + (0.724 + 0.724i)10-s + (0.948 − 0.254i)11-s + (−0.607 − 1.05i)13-s + (−0.0834 − 0.0223i)14-s + (0.0412 − 0.0715i)16-s + (−0.144 + 0.989i)17-s + 0.810i·19-s + (−0.237 − 0.886i)20-s + (0.614 + 0.164i)22-s + (0.323 − 1.20i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(459\)    =    \(3^{3} \cdot 17\)
Sign: $0.996 + 0.0856i$
Analytic conductor: \(3.66513\)
Root analytic conductor: \(1.91445\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{459} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 459,\ (\ :1/2),\ 0.996 + 0.0856i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17575 - 0.0933474i\)
\(L(\frac12)\) \(\approx\) \(2.17575 - 0.0933474i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 + (0.593 - 4.08i)T \)
good2 \( 1 + (-0.793 - 0.458i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-3.41 - 0.915i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.340 - 0.0913i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3.14 + 0.843i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (2.19 + 3.79i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 - 3.53iT - 19T^{2} \)
23 \( 1 + (-1.54 + 5.78i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.880 - 3.28i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-6.39 - 1.71i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.16 - 1.16i)T - 37iT^{2} \)
41 \( 1 + (-1.59 + 5.95i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.98 + 2.87i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.45 - 2.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 11.0iT - 53T^{2} \)
59 \( 1 + (-2.82 + 1.62i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.30 - 1.15i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (5.68 + 9.84i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.98 - 3.98i)T - 71iT^{2} \)
73 \( 1 + (10.0 - 10.0i)T - 73iT^{2} \)
79 \( 1 + (10.7 - 2.88i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (3.56 + 2.05i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 1.04T + 89T^{2} \)
97 \( 1 + (-1.26 - 4.73i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.58451675297791856152567279518, −10.27474465999293700907945852064, −9.422383637398046851902062056465, −8.485008909086342260566254969802, −6.89277274403206866523261800611, −6.15935656768773145429534765344, −5.59997283986505260155478654743, −4.44930020511182834720561086163, −3.01707269408466727180322586996, −1.45585544602072140026150270703, 1.79363443633336014439636301581, 2.92413384414537863941989564852, 4.43682922561133898465165192739, 5.09360205898992430058452561122, 6.29056125133981738893007985250, 7.22653159007599022551830687936, 8.665747831704375597574333569076, 9.428307670877771941860172745151, 9.857915720300607951541773266275, 11.56501271069805686489849497478

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