Properties

Label 2-15e2-225.121-c1-0-16
Degree $2$
Conductor $225$
Sign $0.996 + 0.0872i$
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  + (2.14 − 0.456i)2-s + (−1.28 + 1.15i)3-s + (2.57 − 1.14i)4-s + (0.900 − 2.04i)5-s + (−2.23 + 3.07i)6-s + (1.61 + 2.79i)7-s + (1.45 − 1.05i)8-s + (0.310 − 2.98i)9-s + (0.998 − 4.80i)10-s + (5.10 − 1.08i)11-s + (−1.98 + 4.46i)12-s + (−3.76 − 0.800i)13-s + (4.73 + 5.26i)14-s + (1.21 + 3.67i)15-s + (−1.12 + 1.25i)16-s + (−3.55 + 2.58i)17-s + ⋯
L(s)  = 1  + (1.51 − 0.322i)2-s + (−0.742 + 0.669i)3-s + (1.28 − 0.573i)4-s + (0.402 − 0.915i)5-s + (−0.911 + 1.25i)6-s + (0.609 + 1.05i)7-s + (0.515 − 0.374i)8-s + (0.103 − 0.994i)9-s + (0.315 − 1.52i)10-s + (1.54 − 0.327i)11-s + (−0.572 + 1.28i)12-s + (−1.04 − 0.222i)13-s + (1.26 + 1.40i)14-s + (0.313 + 0.949i)15-s + (−0.281 + 0.312i)16-s + (−0.863 + 0.627i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.996 + 0.0872i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.996 + 0.0872i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27388 - 0.0994159i\)
\(L(\frac12)\) \(\approx\) \(2.27388 - 0.0994159i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.28 - 1.15i)T \)
5 \( 1 + (-0.900 + 2.04i)T \)
good2 \( 1 + (-2.14 + 0.456i)T + (1.82 - 0.813i)T^{2} \)
7 \( 1 + (-1.61 - 2.79i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.10 + 1.08i)T + (10.0 - 4.47i)T^{2} \)
13 \( 1 + (3.76 + 0.800i)T + (11.8 + 5.28i)T^{2} \)
17 \( 1 + (3.55 - 2.58i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.66 - 1.93i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (5.59 + 6.21i)T + (-2.40 + 22.8i)T^{2} \)
29 \( 1 + (0.104 - 0.997i)T + (-28.3 - 6.02i)T^{2} \)
31 \( 1 + (0.352 + 3.35i)T + (-30.3 + 6.44i)T^{2} \)
37 \( 1 + (0.878 + 2.70i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-5.99 - 1.27i)T + (37.4 + 16.6i)T^{2} \)
43 \( 1 + (-2.26 - 3.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.111 - 1.05i)T + (-45.9 - 9.77i)T^{2} \)
53 \( 1 + (-6.03 - 4.38i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.06 + 0.652i)T + (53.8 + 23.9i)T^{2} \)
61 \( 1 + (8.43 - 1.79i)T + (55.7 - 24.8i)T^{2} \)
67 \( 1 + (0.364 + 3.46i)T + (-65.5 + 13.9i)T^{2} \)
71 \( 1 + (-11.0 - 8.01i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.320 - 0.986i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-0.840 + 7.99i)T + (-77.2 - 16.4i)T^{2} \)
83 \( 1 + (-5.24 - 2.33i)T + (55.5 + 61.6i)T^{2} \)
89 \( 1 + (-2.92 + 9.00i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-0.199 + 1.90i)T + (-94.8 - 20.1i)T^{2} \)
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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

−12.22287296434191793132424757399, −11.74865725706083012199306750631, −10.70040242064838970247243886305, −9.366238380431492626746705821548, −8.567545949350684862232745733782, −6.28426023296866765330037878618, −5.81578307644629589156850720284, −4.66901947058869802847841306267, −4.10941849847059597579346532024, −2.13554379682787695366777373556, 2.10541673535744070198756469965, 3.95064937085860692157698545276, 4.88445708197733768837331502126, 6.16119148341291015256945111408, 6.94015997193926068006829988632, 7.42286980099417290376281710445, 9.562175590367867301888169106335, 10.84503343967159606331470786142, 11.61533815238377081018325997488, 12.25408461003072792383984251454

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