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Properties

Label 2-15e2-5.3-c2-0-6
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  + 4i·4-s + (6.12 − 6.12i)7-s + (18.3 + 18.3i)13-s − 16·16-s + 37i·19-s + (24.4 + 24.4i)28-s + 13·31-s + (48.9 − 48.9i)37-s + (−42.8 − 42.8i)43-s − 26i·49-s + (−73.4 + 73.4i)52-s + 47·61-s − 64i·64-s + (−55.1 + 55.1i)67-s + (−97.9 − 97.9i)73-s + ⋯
L(s)  = 1  + i·4-s + (0.874 − 0.874i)7-s + (1.41 + 1.41i)13-s − 16-s + 1.94i·19-s + (0.874 + 0.874i)28-s + 0.419·31-s + (1.32 − 1.32i)37-s + (−0.996 − 0.996i)43-s − 0.530i·49-s + (−1.41 + 1.41i)52-s + 0.770·61-s − i·64-s + (−0.822 + 0.822i)67-s + (−1.34 − 1.34i)73-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: $\chi_{225} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1),\ 0.608 - 0.793i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.52940 + 0.754453i\)
\(L(\frac12)\) \(\approx\) \(1.52940 + 0.754453i\)
\(L(2)\) 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 \( 1 - 4iT^{2} \)
7 \( 1 + (-6.12 + 6.12i)T - 49iT^{2} \)
11 \( 1 + 121T^{2} \)
13 \( 1 + (-18.3 - 18.3i)T + 169iT^{2} \)
17 \( 1 - 289iT^{2} \)
19 \( 1 - 37iT - 361T^{2} \)
23 \( 1 + 529iT^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 - 13T + 961T^{2} \)
37 \( 1 + (-48.9 + 48.9i)T - 1.36e3iT^{2} \)
41 \( 1 + 1.68e3T^{2} \)
43 \( 1 + (42.8 + 42.8i)T + 1.84e3iT^{2} \)
47 \( 1 - 2.20e3iT^{2} \)
53 \( 1 + 2.80e3iT^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 - 47T + 3.72e3T^{2} \)
67 \( 1 + (55.1 - 55.1i)T - 4.48e3iT^{2} \)
71 \( 1 + 5.04e3T^{2} \)
73 \( 1 + (97.9 + 97.9i)T + 5.32e3iT^{2} \)
79 \( 1 + 142iT - 6.24e3T^{2} \)
83 \( 1 + 6.88e3iT^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + (-67.3 + 67.3i)T - 9.40e3iT^{2} \)
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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

−11.99080404864843652717816924119, −11.35602992382191588380815137763, −10.39289472059488422718435270236, −8.988999610558165170394078475791, −8.142171385826094989718768809993, −7.32132949113411951999078432820, −6.12525509673938296103959745486, −4.36338157308747118081352390111, −3.69473738151865981957809564848, −1.68749466644631709000146762114, 1.09268844859616997300644226410, 2.73995763901192192210472662713, 4.71982615641361271262246974519, 5.58323240504541480017343384588, 6.54463281342431248781369248283, 8.121907735713472106111504316050, 8.879284116567907814526998344346, 10.00013119634411952392041342719, 11.08733197830568878874939104636, 11.49442388404282552959043392896

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