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Properties

Label 2-15e2-5.2-c2-0-0
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  + (−2.44 − 2.44i)2-s + 7.99i·4-s + (−6.12 − 6.12i)7-s + (9.79 − 9.79i)8-s − 6·11-s + (−3.67 + 3.67i)13-s + 29.9i·14-s − 15.9·16-s + (17.1 + 17.1i)17-s + 23i·19-s + (14.6 + 14.6i)22-s + (−12.2 + 12.2i)23-s + 18·26-s + (48.9 − 48.9i)28-s − 6i·29-s + ⋯
L(s)  = 1  + (−1.22 − 1.22i)2-s + 1.99i·4-s + (−0.874 − 0.874i)7-s + (1.22 − 1.22i)8-s − 0.545·11-s + (−0.282 + 0.282i)13-s + 2.14i·14-s − 0.999·16-s + (1.00 + 1.00i)17-s + 1.21i·19-s + (0.668 + 0.668i)22-s + (−0.532 + 0.532i)23-s + 0.692·26-s + (1.74 − 1.74i)28-s − 0.206i·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

\(L(\frac{3}{2})\) \(\approx\) \(0.397883 + 0.0916720i\)
\(L(\frac12)\) \(\approx\) \(0.397883 + 0.0916720i\)
\(L(2)\) 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 \( 1 + (2.44 + 2.44i)T + 4iT^{2} \)
7 \( 1 + (6.12 + 6.12i)T + 49iT^{2} \)
11 \( 1 + 6T + 121T^{2} \)
13 \( 1 + (3.67 - 3.67i)T - 169iT^{2} \)
17 \( 1 + (-17.1 - 17.1i)T + 289iT^{2} \)
19 \( 1 - 23iT - 361T^{2} \)
23 \( 1 + (12.2 - 12.2i)T - 529iT^{2} \)
29 \( 1 + 6iT - 841T^{2} \)
31 \( 1 - 25T + 961T^{2} \)
37 \( 1 + (-24.4 - 24.4i)T + 1.36e3iT^{2} \)
41 \( 1 - 60T + 1.68e3T^{2} \)
43 \( 1 + (60.0 - 60.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (-7.34 - 7.34i)T + 2.20e3iT^{2} \)
53 \( 1 + (24.4 - 24.4i)T - 2.80e3iT^{2} \)
59 \( 1 - 18iT - 3.48e3T^{2} \)
61 \( 1 + 37T + 3.72e3T^{2} \)
67 \( 1 + (-25.7 - 25.7i)T + 4.48e3iT^{2} \)
71 \( 1 + 132T + 5.04e3T^{2} \)
73 \( 1 + (-24.4 + 24.4i)T - 5.32e3iT^{2} \)
79 \( 1 - 10iT - 6.24e3T^{2} \)
83 \( 1 + (-2.44 + 2.44i)T - 6.88e3iT^{2} \)
89 \( 1 - 132iT - 7.92e3T^{2} \)
97 \( 1 + (23.2 + 23.2i)T + 9.40e3iT^{2} \)
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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

−11.95080217091876383788188882245, −10.83363565353012034063279755313, −10.01192818797533133100536364712, −9.686119511052546663133007036293, −8.216763257625378124914596223649, −7.60121954922206006590605326142, −6.10296152920069319670427363293, −4.02695098136024601350851357510, −2.98303909394624615090210818392, −1.33275656866598676725437277993, 0.35755044013390453147128502600, 2.75857922997806676076361024757, 5.12807853554647832557193359186, 6.04219386293400109331606502674, 7.04346289059040179539233832858, 7.952023197328658135539066912349, 8.990150995689655074835520348480, 9.643761082990059142686505264525, 10.48794862713736806217072764540, 11.88312934232206017352618292328

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