Properties

Label 2-15e2-5.2-c2-0-13
Degree $2$
Conductor $225$
Sign $-0.608 - 0.793i$
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  + (−1.22 − 1.22i)2-s − 1.00i·4-s + (−4.89 − 4.89i)7-s + (−6.12 + 6.12i)8-s + 3·11-s + (−7.34 + 7.34i)13-s + 11.9i·14-s + 10.9·16-s + (−13.4 − 13.4i)17-s + 5i·19-s + (−3.67 − 3.67i)22-s + (−17.1 + 17.1i)23-s + 18·26-s + (−4.89 + 4.89i)28-s + 30i·29-s + ⋯
L(s)  = 1  + (−0.612 − 0.612i)2-s − 0.250i·4-s + (−0.699 − 0.699i)7-s + (−0.765 + 0.765i)8-s + 0.272·11-s + (−0.565 + 0.565i)13-s + 0.857i·14-s + 0.687·16-s + (−0.792 − 0.792i)17-s + 0.263i·19-s + (−0.167 − 0.167i)22-s + (−0.745 + 0.745i)23-s + 0.692·26-s + (−0.174 + 0.174i)28-s + 1.03i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.608 - 0.793i$
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: \((2,\ 225,\ (\ :1),\ -0.608 - 0.793i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0597231 + 0.121069i\)
\(L(\frac12)\) \(\approx\) \(0.0597231 + 0.121069i\)
\(L(2)\) 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 \( 1 + (1.22 + 1.22i)T + 4iT^{2} \)
7 \( 1 + (4.89 + 4.89i)T + 49iT^{2} \)
11 \( 1 - 3T + 121T^{2} \)
13 \( 1 + (7.34 - 7.34i)T - 169iT^{2} \)
17 \( 1 + (13.4 + 13.4i)T + 289iT^{2} \)
19 \( 1 - 5iT - 361T^{2} \)
23 \( 1 + (17.1 - 17.1i)T - 529iT^{2} \)
29 \( 1 - 30iT - 841T^{2} \)
31 \( 1 + 38T + 961T^{2} \)
37 \( 1 + (-19.5 - 19.5i)T + 1.36e3iT^{2} \)
41 \( 1 + 57T + 1.68e3T^{2} \)
43 \( 1 + (-4.89 + 4.89i)T - 1.84e3iT^{2} \)
47 \( 1 + (7.34 + 7.34i)T + 2.20e3iT^{2} \)
53 \( 1 + (-31.8 + 31.8i)T - 2.80e3iT^{2} \)
59 \( 1 + 90iT - 3.48e3T^{2} \)
61 \( 1 + 28T + 3.72e3T^{2} \)
67 \( 1 + (47.7 + 47.7i)T + 4.48e3iT^{2} \)
71 \( 1 + 42T + 5.04e3T^{2} \)
73 \( 1 + (13.4 - 13.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 80iT - 6.24e3T^{2} \)
83 \( 1 + (-111. + 111. i)T - 6.88e3iT^{2} \)
89 \( 1 - 15iT - 7.92e3T^{2} \)
97 \( 1 + (53.8 + 53.8i)T + 9.40e3iT^{2} \)
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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

−11.29506231130090077524540759744, −10.29639715551975428188743392400, −9.593796146917377975242974361027, −8.816064839565592963096325085293, −7.35171562604295048599959053234, −6.36828837680378856227475322096, −4.97818828669437133914202143160, −3.43481396056161488113640760637, −1.84883495312584071143165209526, −0.084607343842383066068466514975, 2.61123488384729660036919720633, 4.03637687945360938964497067194, 5.80268265136281967277961176080, 6.65053411589934279969974893977, 7.74443982720865783610740897097, 8.710995006050827522679517770718, 9.436142571758504169925928690631, 10.46800180106366545235009215937, 11.86956891383589005223024314565, 12.58546008069906418545767082114

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