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Properties

Label 2-15e2-5.2-c2-0-11
Degree $2$
Conductor $225$
Sign $-0.525 + 0.850i$
Analytic cond. $6.13080$
Root an. cond. $2.47604$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.224 + 0.224i)2-s − 3.89i·4-s + (−3.44 − 3.44i)7-s + (1.77 − 1.77i)8-s − 11.3·11-s + (5.55 − 5.55i)13-s − 1.55i·14-s − 14.7·16-s + (−17.3 − 17.3i)17-s + 8.69i·19-s + (−2.55 − 2.55i)22-s + (11.5 − 11.5i)23-s + 2.49·26-s + (−13.4 + 13.4i)28-s − 35.1i·29-s + ⋯
L(s)  = 1  + (0.112 + 0.112i)2-s − 0.974i·4-s + (−0.492 − 0.492i)7-s + (0.221 − 0.221i)8-s − 1.03·11-s + (0.426 − 0.426i)13-s − 0.110i·14-s − 0.924·16-s + (−1.02 − 1.02i)17-s + 0.457i·19-s + (−0.115 − 0.115i)22-s + (0.502 − 0.502i)23-s + 0.0959·26-s + (−0.480 + 0.480i)28-s − 1.21i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.525 + 0.850i$
Analytic conductor: \(6.13080\)
Root analytic conductor: \(2.47604\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1),\ -0.525 + 0.850i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.535386 - 0.960272i\)
\(L(\frac12)\) \(\approx\) \(0.535386 - 0.960272i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.224 - 0.224i)T + 4iT^{2} \)
7 \( 1 + (3.44 + 3.44i)T + 49iT^{2} \)
11 \( 1 + 11.3T + 121T^{2} \)
13 \( 1 + (-5.55 + 5.55i)T - 169iT^{2} \)
17 \( 1 + (17.3 + 17.3i)T + 289iT^{2} \)
19 \( 1 - 8.69iT - 361T^{2} \)
23 \( 1 + (-11.5 + 11.5i)T - 529iT^{2} \)
29 \( 1 + 35.1iT - 841T^{2} \)
31 \( 1 - 10.6T + 961T^{2} \)
37 \( 1 + (-6.04 - 6.04i)T + 1.36e3iT^{2} \)
41 \( 1 + 0.696T + 1.68e3T^{2} \)
43 \( 1 + (-26.4 + 26.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (-44.2 - 44.2i)T + 2.20e3iT^{2} \)
53 \( 1 + (0.696 - 0.696i)T - 2.80e3iT^{2} \)
59 \( 1 - 39.9iT - 3.48e3T^{2} \)
61 \( 1 - 5.90T + 3.72e3T^{2} \)
67 \( 1 + (-45.1 - 45.1i)T + 4.48e3iT^{2} \)
71 \( 1 - 68T + 5.04e3T^{2} \)
73 \( 1 + (77.7 - 77.7i)T - 5.32e3iT^{2} \)
79 \( 1 + 24.4iT - 6.24e3T^{2} \)
83 \( 1 + (-13.1 + 13.1i)T - 6.88e3iT^{2} \)
89 \( 1 + 82.1iT - 7.92e3T^{2} \)
97 \( 1 + (-24.5 - 24.5i)T + 9.40e3iT^{2} \)
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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

−11.49679743513310089915117842575, −10.58517369895223074999926546395, −9.950508808261636602058759166009, −8.857511799366143256330519183974, −7.52257805448851540024171845167, −6.48374007198794133654159174354, −5.46132822901386991078901940701, −4.32402725310536356477860026764, −2.56426142020865086970385653538, −0.55708006345420739640671132517, 2.36140027017480890621960571235, 3.55654715797918749650014039214, 4.86716117192925319806462745489, 6.29239232322289397479535755583, 7.37966063108100749050732323340, 8.486885153823864915675400631952, 9.184783549582935469198659225124, 10.63673146277486606482652384171, 11.39476394926789808720723258612, 12.58394605085179503213649033274

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