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Properties

Label 2-15e2-15.2-c3-0-2
Degree $2$
Conductor $225$
Sign $-0.998 + 0.0618i$
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  + (−2.62 + 2.62i)2-s − 5.74i·4-s + (10.8 + 10.8i)7-s + (−5.91 − 5.91i)8-s + 37.8i·11-s + (48.1 − 48.1i)13-s − 56.9·14-s + 76.9·16-s + (−60.3 + 60.3i)17-s + 109. i·19-s + (−99.3 − 99.3i)22-s + (39.7 + 39.7i)23-s + 252. i·26-s + (62.4 − 62.4i)28-s − 90.3·29-s + ⋯
L(s)  = 1  + (−0.926 + 0.926i)2-s − 0.717i·4-s + (0.586 + 0.586i)7-s + (−0.261 − 0.261i)8-s + 1.03i·11-s + (1.02 − 1.02i)13-s − 1.08·14-s + 1.20·16-s + (−0.860 + 0.860i)17-s + 1.32i·19-s + (−0.962 − 0.962i)22-s + (0.360 + 0.360i)23-s + 1.90i·26-s + (0.421 − 0.421i)28-s − 0.578·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.998 + 0.0618i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.998 + 0.0618i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0229912 - 0.742549i\)
\(L(\frac12)\) \(\approx\) \(0.0229912 - 0.742549i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.62 - 2.62i)T - 8iT^{2} \)
7 \( 1 + (-10.8 - 10.8i)T + 343iT^{2} \)
11 \( 1 - 37.8iT - 1.33e3T^{2} \)
13 \( 1 + (-48.1 + 48.1i)T - 2.19e3iT^{2} \)
17 \( 1 + (60.3 - 60.3i)T - 4.91e3iT^{2} \)
19 \( 1 - 109. iT - 6.85e3T^{2} \)
23 \( 1 + (-39.7 - 39.7i)T + 1.21e4iT^{2} \)
29 \( 1 + 90.3T + 2.43e4T^{2} \)
31 \( 1 + 233.T + 2.97e4T^{2} \)
37 \( 1 + (-19.0 - 19.0i)T + 5.06e4iT^{2} \)
41 \( 1 + 260. iT - 6.89e4T^{2} \)
43 \( 1 + (176. - 176. i)T - 7.95e4iT^{2} \)
47 \( 1 + (145. - 145. i)T - 1.03e5iT^{2} \)
53 \( 1 + (183. + 183. i)T + 1.48e5iT^{2} \)
59 \( 1 + 279.T + 2.05e5T^{2} \)
61 \( 1 + 390.T + 2.26e5T^{2} \)
67 \( 1 + (150. + 150. i)T + 3.00e5iT^{2} \)
71 \( 1 - 470. iT - 3.57e5T^{2} \)
73 \( 1 + (480. - 480. i)T - 3.89e5iT^{2} \)
79 \( 1 - 1.32e3iT - 4.93e5T^{2} \)
83 \( 1 + (-456. - 456. i)T + 5.71e5iT^{2} \)
89 \( 1 - 1.36e3T + 7.04e5T^{2} \)
97 \( 1 + (-785. - 785. i)T + 9.12e5iT^{2} \)
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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

−12.39378698020970895603277076522, −11.09113175605204959912924427678, −10.12125922911786110077832150238, −9.066808099834375909805010610762, −8.267648070832126186802546430354, −7.54041478676750338513954095516, −6.32206389167076993559426961728, −5.40071426364999466820357840226, −3.69557882700024288901082343816, −1.65952005047355009081023598849, 0.42795333054892905625036077541, 1.74503883455570899848043945379, 3.23738335499337420361420216431, 4.72367946094237753873008408711, 6.26639426082712827095313217470, 7.54238917144381946678434763945, 8.904622070297068770105513720254, 9.083294819546677507077121425282, 10.62337504958390042818092935184, 11.19262914839349022482685028996

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