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Properties

Label 2-15e2-5.2-c2-0-7
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  + (1.22 + 1.22i)2-s − 1.00i·4-s + (4.89 + 4.89i)7-s + (6.12 − 6.12i)8-s + 3·11-s + (7.34 − 7.34i)13-s + 11.9i·14-s + 10.9·16-s + (13.4 + 13.4i)17-s + 5i·19-s + (3.67 + 3.67i)22-s + (17.1 − 17.1i)23-s + 18·26-s + (4.89 − 4.89i)28-s + 30i·29-s + ⋯
L(s)  = 1  + (0.612 + 0.612i)2-s − 0.250i·4-s + (0.699 + 0.699i)7-s + (0.765 − 0.765i)8-s + 0.272·11-s + (0.565 − 0.565i)13-s + 0.857i·14-s + 0.687·16-s + (0.792 + 0.792i)17-s + 0.263i·19-s + (0.167 + 0.167i)22-s + (0.745 − 0.745i)23-s + 0.692·26-s + (0.174 − 0.174i)28-s + 1.03i·29-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: $\chi_{225} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1),\ 0.899 - 0.437i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.34347 + 0.539934i\)
\(L(\frac12)\) \(\approx\) \(2.34347 + 0.539934i\)
\(L(2)\) 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 \( 1 + (-1.22 - 1.22i)T + 4iT^{2} \)
7 \( 1 + (-4.89 - 4.89i)T + 49iT^{2} \)
11 \( 1 - 3T + 121T^{2} \)
13 \( 1 + (-7.34 + 7.34i)T - 169iT^{2} \)
17 \( 1 + (-13.4 - 13.4i)T + 289iT^{2} \)
19 \( 1 - 5iT - 361T^{2} \)
23 \( 1 + (-17.1 + 17.1i)T - 529iT^{2} \)
29 \( 1 - 30iT - 841T^{2} \)
31 \( 1 + 38T + 961T^{2} \)
37 \( 1 + (19.5 + 19.5i)T + 1.36e3iT^{2} \)
41 \( 1 + 57T + 1.68e3T^{2} \)
43 \( 1 + (4.89 - 4.89i)T - 1.84e3iT^{2} \)
47 \( 1 + (-7.34 - 7.34i)T + 2.20e3iT^{2} \)
53 \( 1 + (31.8 - 31.8i)T - 2.80e3iT^{2} \)
59 \( 1 + 90iT - 3.48e3T^{2} \)
61 \( 1 + 28T + 3.72e3T^{2} \)
67 \( 1 + (-47.7 - 47.7i)T + 4.48e3iT^{2} \)
71 \( 1 + 42T + 5.04e3T^{2} \)
73 \( 1 + (-13.4 + 13.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 80iT - 6.24e3T^{2} \)
83 \( 1 + (111. - 111. i)T - 6.88e3iT^{2} \)
89 \( 1 - 15iT - 7.92e3T^{2} \)
97 \( 1 + (-53.8 - 53.8i)T + 9.40e3iT^{2} \)
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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

−12.34736086186204518856581931638, −11.06242637237477068679528908373, −10.30585750009338147021470993588, −8.993812806430919426708364828428, −8.018394320683349624310175932110, −6.80711956296735100158460333586, −5.72655264966527574820639343107, −5.01976207111317051179412876354, −3.58728750946998076273035988448, −1.53185053427783047779234247959, 1.57094550524415515468253757216, 3.24975858714790882141130346495, 4.29146790071985406439510430553, 5.32998019900926768765505391095, 7.02211502905865862740468964520, 7.87686297864578694477561461757, 9.018918614642254109912797955193, 10.30913828669451554409643666451, 11.37242253197311315363200387928, 11.72290190959703315006143495457

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