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Properties

Label 2-15e2-5.4-c3-0-6
Degree $2$
Conductor $225$
Sign $-0.894 - 0.447i$
Analytic cond. $13.2754$
Root an. cond. $3.64354$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.16i·2-s − 2.00·4-s + 15i·7-s + 18.9i·8-s + 63.2·11-s − 35i·13-s − 47.4·14-s − 76·16-s + 88.5i·17-s − 91·19-s + 200i·22-s + 113. i·23-s + 110.·26-s − 30.0i·28-s + 63.2·29-s + ⋯
L(s)  = 1  + 1.11i·2-s − 0.250·4-s + 0.809i·7-s + 0.838i·8-s + 1.73·11-s − 0.746i·13-s − 0.905·14-s − 1.18·16-s + 1.26i·17-s − 1.09·19-s + 1.93i·22-s + 1.03i·23-s + 0.834·26-s − 0.202i·28-s + 0.404·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.422742 + 1.79076i\)
\(L(\frac12)\) \(\approx\) \(0.422742 + 1.79076i\)
\(L(\frac{5}{2})\) 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 \( 1 - 3.16iT - 8T^{2} \)
7 \( 1 - 15iT - 343T^{2} \)
11 \( 1 - 63.2T + 1.33e3T^{2} \)
13 \( 1 + 35iT - 2.19e3T^{2} \)
17 \( 1 - 88.5iT - 4.91e3T^{2} \)
19 \( 1 + 91T + 6.85e3T^{2} \)
23 \( 1 - 113. iT - 1.21e4T^{2} \)
29 \( 1 - 63.2T + 2.43e4T^{2} \)
31 \( 1 + 147T + 2.97e4T^{2} \)
37 \( 1 - 370iT - 5.06e4T^{2} \)
41 \( 1 + 442.T + 6.89e4T^{2} \)
43 \( 1 + 335iT - 7.95e4T^{2} \)
47 \( 1 + 177. iT - 1.03e5T^{2} \)
53 \( 1 + 88.5iT - 1.48e5T^{2} \)
59 \( 1 - 885.T + 2.05e5T^{2} \)
61 \( 1 - 427T + 2.26e5T^{2} \)
67 \( 1 - 15iT - 3.00e5T^{2} \)
71 \( 1 + 63.2T + 3.57e5T^{2} \)
73 \( 1 - 70iT - 3.89e5T^{2} \)
79 \( 1 - 876T + 4.93e5T^{2} \)
83 \( 1 - 531. iT - 5.71e5T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 + 1.08e3iT - 9.12e5T^{2} \)
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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

−12.13735684172279417258124355847, −11.41818295636773680677797920500, −10.15455539564600744698719550290, −8.787138186044000680762388185645, −8.349394066343640826205752344731, −6.94346074939275645248964598673, −6.21144216567498654818943403921, −5.28892256675109539182266452110, −3.72526638631875332885558906131, −1.87105558585267892031174603272, 0.77728245128194881005115747777, 2.10944216568688078139541795858, 3.67673531946017526262660277463, 4.45860339148643725857178659170, 6.51156061203672841131135801078, 7.09600231809642229642619020899, 8.833533742929263749890424989859, 9.606029121416750278604896144477, 10.61446338118057407107614262291, 11.42203763566379165457412928795

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