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Properties

Label 2-15e2-25.11-c1-0-2
Degree $2$
Conductor $225$
Sign $0.327 - 0.944i$
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  + (−1.63 + 1.18i)2-s + (0.641 − 1.97i)4-s + (2.07 + 0.843i)5-s + 1.01·7-s + (0.0473 + 0.145i)8-s + (−4.38 + 1.07i)10-s + (3.85 − 2.79i)11-s + (0.0840 + 0.0610i)13-s + (−1.66 + 1.20i)14-s + (3.10 + 2.25i)16-s + (1.80 + 5.55i)17-s + (−0.223 − 0.688i)19-s + (2.99 − 3.54i)20-s + (−2.96 + 9.13i)22-s + (−7.33 + 5.33i)23-s + ⋯
L(s)  = 1  + (−1.15 + 0.839i)2-s + (0.320 − 0.987i)4-s + (0.926 + 0.377i)5-s + 0.385·7-s + (0.0167 + 0.0514i)8-s + (−1.38 + 0.341i)10-s + (1.16 − 0.843i)11-s + (0.0232 + 0.0169i)13-s + (−0.444 + 0.323i)14-s + (0.777 + 0.564i)16-s + (0.437 + 1.34i)17-s + (−0.0513 − 0.158i)19-s + (0.669 − 0.793i)20-s + (−0.633 + 1.94i)22-s + (−1.53 + 1.11i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.327 - 0.944i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.327 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.693509 + 0.493444i\)
\(L(\frac12)\) \(\approx\) \(0.693509 + 0.493444i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-2.07 - 0.843i)T \)
good2 \( 1 + (1.63 - 1.18i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 - 1.01T + 7T^{2} \)
11 \( 1 + (-3.85 + 2.79i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.0840 - 0.0610i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.80 - 5.55i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.223 + 0.688i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (7.33 - 5.33i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.23 + 3.79i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.329 - 1.01i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.25 + 2.36i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-5.83 - 4.23i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 8.62T + 43T^{2} \)
47 \( 1 + (-2.53 + 7.79i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.34 + 4.15i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.97 + 2.88i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (5.63 - 4.09i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (3.06 + 9.43i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-3.33 + 10.2i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (6.98 - 5.07i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.767 + 2.36i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.31 + 4.03i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (14.8 - 10.8i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-2.07 + 6.37i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.36710875091004882112600654527, −11.13823982336492055232228523676, −10.16947734928938593216038712168, −9.379929212968957235838413463761, −8.511953173276536029166834632806, −7.58471697905861177592271417257, −6.30846590231575462049348152699, −5.85555628188951764276816499018, −3.74508412511464790025259479610, −1.56140787561405278751136635303, 1.30823927321469147368419195177, 2.51046993396445143284950057804, 4.50934526421255859698811692025, 5.90144274086218456582418916163, 7.29843493549267592546194639802, 8.528346914599518264315719091564, 9.365062529634747395173916106575, 9.924617620124054744574802942762, 10.87622852602852412814018362122, 12.00863278772722878060341135499

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