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Properties

Label 2-15e2-9.4-c3-0-1
Degree $2$
Conductor $225$
Sign $-0.293 - 0.955i$
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  + (−2.77 − 4.80i)2-s + (0.320 + 5.18i)3-s + (−11.3 + 19.6i)4-s + (24.0 − 15.9i)6-s + (12.8 + 22.2i)7-s + 81.7·8-s + (−26.7 + 3.32i)9-s + (−3.12 − 5.41i)11-s + (−105. − 52.6i)12-s + (−7.62 + 13.2i)13-s + (71.2 − 123. i)14-s + (−135. − 234. i)16-s + 36.0·17-s + (90.2 + 119. i)18-s − 52.7·19-s + ⋯
L(s)  = 1  + (−0.980 − 1.69i)2-s + (0.0617 + 0.998i)3-s + (−1.42 + 2.46i)4-s + (1.63 − 1.08i)6-s + (0.694 + 1.20i)7-s + 3.61·8-s + (−0.992 + 0.123i)9-s + (−0.0857 − 0.148i)11-s + (−2.54 − 1.26i)12-s + (−0.162 + 0.281i)13-s + (1.36 − 2.35i)14-s + (−2.11 − 3.66i)16-s + 0.513·17-s + (1.18 + 1.56i)18-s − 0.636·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.293 - 0.955i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.293 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.293 - 0.955i$
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: \((2,\ 225,\ (\ :3/2),\ -0.293 - 0.955i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.273186 + 0.369726i\)
\(L(\frac12)\) \(\approx\) \(0.273186 + 0.369726i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.320 - 5.18i)T \)
5 \( 1 \)
good2 \( 1 + (2.77 + 4.80i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (-12.8 - 22.2i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (3.12 + 5.41i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (7.62 - 13.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 36.0T + 4.91e3T^{2} \)
19 \( 1 + 52.7T + 6.85e3T^{2} \)
23 \( 1 + (41.8 - 72.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (59.5 + 103. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (138. - 239. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 117.T + 5.06e4T^{2} \)
41 \( 1 + (-79.6 + 137. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (147. + 255. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-41.6 - 72.1i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 149.T + 1.48e5T^{2} \)
59 \( 1 + (317. - 550. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (298. + 517. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (165. - 285. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 1.14e3T + 3.57e5T^{2} \)
73 \( 1 - 130.T + 3.89e5T^{2} \)
79 \( 1 + (-368. - 638. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (184. + 320. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 225.T + 7.04e5T^{2} \)
97 \( 1 + (5.95 + 10.3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.78355275124548271586605562882, −11.02892866914423396794897591671, −10.22987976926594736750650649816, −9.248034734887086112746188109533, −8.724392635177377170491983231008, −7.80971535917936704636448377658, −5.38999620897243973556301005325, −4.20646898431641580139289385823, −3.01026963224603153281450362524, −1.87713769103400310379957555196, 0.27225336611105279389967589540, 1.50914041399595359352240727010, 4.49821215871499264767584235993, 5.76655114871595234830408129505, 6.71995764085410337782100421144, 7.67630624057151879050972878037, 7.973559403735871024983777631036, 9.152226279413036269945159031051, 10.29869031175600367330431900346, 11.15859455980836986994905271059

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