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Properties

Label 2-15e2-5.3-c2-0-2
Degree $2$
Conductor $225$
Sign $-0.899 - 0.437i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.437i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.899 - 0.437i$
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: \((2,\ 225,\ (\ :1),\ -0.899 - 0.437i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.169959 + 0.737671i\)
\(L(\frac12)\) \(\approx\) \(0.169959 + 0.737671i\)
\(L(2)\) 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

−12.44093331115906064977419655687, −11.87803151246391376590181245452, −10.33240942002039448740937336754, −9.548887567408869062186108031711, −8.337009814233681948363022096054, −7.62317130757821839450442928037, −6.33456026987217426011316657894, −5.15783637662782680491300358999, −3.53181649080284101719209177586, −2.57744017340251332822836556338, 0.38727563138447122473997950474, 2.33687282911949475481194264279, 4.18724580958307321321664442767, 5.22791570800650475067459139324, 6.78514664316896286720266661071, 7.07007966375889320715472572125, 9.056046401586444530865729495868, 9.640034547554890507402984280370, 10.55560915401318119300069626176, 11.45615145543575474832697941561

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