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Properties

Label 2-15e2-9.7-c3-0-6
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: $\chi_{225} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.293 - 0.955i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.413287 + 0.305372i\)
\(L(\frac12)\) \(\approx\) \(0.413287 + 0.305372i\)
\(L(\frac{5}{2})\) 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

−11.88408412234072527470596459262, −11.15934380336843707482988560450, −10.32840771308735584337059624223, −9.285532969341282381904984967554, −8.964733298859284800412875808978, −6.12742115484450565370466955805, −5.37112876578988524730516924887, −4.25746465217846788836206118531, −3.22927883503625286679734765036, −2.18172186960903455697200470698, 0.14585111443996357599321898183, 3.13656598219883568616755231129, 4.34245112604866919399221601710, 5.68757516425089816044505748356, 6.63964636029008977659459814002, 7.16669498435109386229899806781, 8.108939364929335372101366279444, 9.054035756965855822261201223765, 10.82663589725446866879656426652, 12.29227379032109464539729980189

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