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Properties

Label 2-15e2-9.4-c1-0-7
Degree $2$
Conductor $225$
Sign $0.642 + 0.766i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 + 0.448i)2-s + (−1.67 + 0.448i)3-s + (0.866 − 1.5i)4-s + (−0.633 − 0.633i)6-s + (−1.67 − 2.89i)7-s + 1.93·8-s + (2.59 − 1.50i)9-s + (−0.633 − 1.09i)11-s + (−0.776 + 2.89i)12-s + (1.22 − 2.12i)13-s + (0.866 − 1.5i)14-s + (−1.23 − 2.13i)16-s + 5.27·17-s + (1.34 + 0.776i)18-s − 0.732·19-s + ⋯
L(s)  = 1  + (0.183 + 0.316i)2-s + (−0.965 + 0.258i)3-s + (0.433 − 0.750i)4-s + (−0.258 − 0.258i)6-s + (−0.632 − 1.09i)7-s + 0.683·8-s + (0.866 − 0.5i)9-s + (−0.191 − 0.331i)11-s + (−0.224 + 0.836i)12-s + (0.339 − 0.588i)13-s + (0.231 − 0.400i)14-s + (−0.308 − 0.533i)16-s + 1.28·17-s + (0.316 + 0.183i)18-s − 0.167·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.941642 - 0.439095i\)
\(L(\frac12)\) \(\approx\) \(0.941642 - 0.439095i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.67 - 0.448i)T \)
5 \( 1 \)
good2 \( 1 + (-0.258 - 0.448i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (1.67 + 2.89i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.633 + 1.09i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.22 + 2.12i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.27T + 17T^{2} \)
19 \( 1 + 0.732T + 19T^{2} \)
23 \( 1 + (0.258 - 0.448i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.232 - 0.401i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.366 - 0.633i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.24T + 37T^{2} \)
41 \( 1 + (3.86 - 6.69i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.328 - 0.568i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.48 - 2.56i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.03T + 53T^{2} \)
59 \( 1 + (4.73 - 8.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.33 - 5.76i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.79 + 6.57i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 - 8.48T + 73T^{2} \)
79 \( 1 + (3.73 + 6.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.98 - 6.90i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + (-7.58 - 13.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.02061709350806873094563878803, −10.88669051789923960308758778836, −10.37886340281397160254428840031, −9.637999856038884310281868308656, −7.76490125957180324249091261610, −6.78346510483546293470386999336, −5.94354863146864570342915090340, −4.99231730926276296224480984720, −3.57776446103860760857071855834, −0.978511415818909571962898399012, 2.08735180492674634089133573452, 3.61742932060607815507210780877, 5.14365787047990759662960074044, 6.25187794248158054890019236851, 7.18196247613503587788951852499, 8.327427608452564751754158335632, 9.651051651822167104390622979407, 10.71295243057523758962355312241, 11.70204458013232746993843570177, 12.31092779762349648289070087114

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