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Properties

Label 2-15e2-5.4-c3-0-13
Degree $2$
Conductor $225$
Sign $0.894 - 0.447i$
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  + 5.35i·2-s − 20.7·4-s − 4.43i·7-s − 68.1i·8-s + 3.43·11-s − 78.7i·13-s + 23.7·14-s + 199.·16-s − 53.1i·17-s − 20.4·19-s + 18.4i·22-s + 118. i·23-s + 421.·26-s + 91.8i·28-s + 168.·29-s + ⋯
L(s)  = 1  + 1.89i·2-s − 2.58·4-s − 0.239i·7-s − 3.01i·8-s + 0.0941·11-s − 1.67i·13-s + 0.453·14-s + 3.11·16-s − 0.758i·17-s − 0.246·19-s + 0.178i·22-s + 1.07i·23-s + 3.18·26-s + 0.620i·28-s + 1.07·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.949884 + 0.224237i\)
\(L(\frac12)\) \(\approx\) \(0.949884 + 0.224237i\)
\(L(\frac{5}{2})\) 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 \( 1 - 5.35iT - 8T^{2} \)
7 \( 1 + 4.43iT - 343T^{2} \)
11 \( 1 - 3.43T + 1.33e3T^{2} \)
13 \( 1 + 78.7iT - 2.19e3T^{2} \)
17 \( 1 + 53.1iT - 4.91e3T^{2} \)
19 \( 1 + 20.4T + 6.85e3T^{2} \)
23 \( 1 - 118. iT - 1.21e4T^{2} \)
29 \( 1 - 168.T + 2.43e4T^{2} \)
31 \( 1 + 61.0T + 2.97e4T^{2} \)
37 \( 1 + 246. iT - 5.06e4T^{2} \)
41 \( 1 + 422.T + 6.89e4T^{2} \)
43 \( 1 + 362. iT - 7.95e4T^{2} \)
47 \( 1 - 170. iT - 1.03e5T^{2} \)
53 \( 1 + 546. iT - 1.48e5T^{2} \)
59 \( 1 + 216.T + 2.05e5T^{2} \)
61 \( 1 - 130.T + 2.26e5T^{2} \)
67 \( 1 + 614. iT - 3.00e5T^{2} \)
71 \( 1 + 324.T + 3.57e5T^{2} \)
73 \( 1 - 88.8iT - 3.89e5T^{2} \)
79 \( 1 - 1.13e3T + 4.93e5T^{2} \)
83 \( 1 + 758. iT - 5.71e5T^{2} \)
89 \( 1 - 195.T + 7.04e5T^{2} \)
97 \( 1 - 521iT - 9.12e5T^{2} \)
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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

−12.23121114694583489255917825624, −10.55658960112043184698743327739, −9.577293405979104298973087674804, −8.538017321671135534020179315084, −7.70418956734380704437232279788, −6.90087421456586113881046906641, −5.72124477964164700179138551987, −4.97721268861969391748458586616, −3.53605949193784270753537104048, −0.44271855522955452617559092055, 1.41909186164965269806677811753, 2.57511246785799664603667742016, 3.95869801497453443313818730936, 4.81389652396368263983130818412, 6.48173484653766664869584686797, 8.368071932775210930947216354882, 9.058887260239526243841607637786, 10.04318153461359327077473369855, 10.83855705311154688492889704157, 11.81027719846880356459507140254

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