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Properties

Label 2-15e2-9.4-c3-0-37
Degree $2$
Conductor $225$
Sign $-0.767 - 0.641i$
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  + (−0.832 − 1.44i)2-s + (−4.88 + 1.78i)3-s + (2.61 − 4.52i)4-s + (6.63 + 5.55i)6-s + (5.28 + 9.15i)7-s − 22.0·8-s + (20.6 − 17.3i)9-s + (−14.5 − 25.1i)11-s + (−4.69 + 26.7i)12-s + (11.7 − 20.3i)13-s + (8.79 − 15.2i)14-s + (−2.59 − 4.49i)16-s − 35.6·17-s + (−42.2 − 15.2i)18-s + 23.7·19-s + ⋯
L(s)  = 1  + (−0.294 − 0.509i)2-s + (−0.939 + 0.342i)3-s + (0.326 − 0.566i)4-s + (0.451 + 0.377i)6-s + (0.285 + 0.494i)7-s − 0.973·8-s + (0.764 − 0.644i)9-s + (−0.398 − 0.690i)11-s + (−0.112 + 0.643i)12-s + (0.250 − 0.434i)13-s + (0.167 − 0.290i)14-s + (−0.0405 − 0.0702i)16-s − 0.508·17-s + (−0.553 − 0.200i)18-s + 0.286·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 - 0.641i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.767 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.767 - 0.641i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.767 - 0.641i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0342081 + 0.0942584i\)
\(L(\frac12)\) \(\approx\) \(0.0342081 + 0.0942584i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.88 - 1.78i)T \)
5 \( 1 \)
good2 \( 1 + (0.832 + 1.44i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (-5.28 - 9.15i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (14.5 + 25.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-11.7 + 20.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 35.6T + 4.91e3T^{2} \)
19 \( 1 - 23.7T + 6.85e3T^{2} \)
23 \( 1 + (39.4 - 68.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (74.0 + 128. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (170. - 296. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 337.T + 5.06e4T^{2} \)
41 \( 1 + (177. - 306. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (9.76 + 16.9i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (56.0 + 97.1i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 699.T + 1.48e5T^{2} \)
59 \( 1 + (13.7 - 23.7i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-412. - 714. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-412. + 714. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 337.T + 3.57e5T^{2} \)
73 \( 1 - 717.T + 3.89e5T^{2} \)
79 \( 1 + (-155. - 268. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (276. + 478. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.08e3T + 7.04e5T^{2} \)
97 \( 1 + (583. + 1.01e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.15683642990992958490445803037, −10.45349689217840806134417500239, −9.548310877111836549932344511690, −8.481287672041003379354191419660, −6.89198070522722601503837594473, −5.80287266639269607344480915666, −5.13611064171700069772791004909, −3.33181144446870268643323009446, −1.60096155202777425941123880437, −0.04946719098052397155287753449, 1.99780744484573454426733092984, 3.98109978107763138577506621569, 5.31193882361754668883668831481, 6.57002330449972392152587405081, 7.24850472844864329130791577265, 8.105348270824802889747835079978, 9.411134395958237847766692564875, 10.68600110851642956979301026071, 11.40169925686999105882668988310, 12.37394479171243911337705333641

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