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Properties

Label 2-15e2-45.4-c1-0-15
Degree $2$
Conductor $225$
Sign $-0.335 + 0.941i$
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  + (2.17 − 1.25i)2-s + (−1.19 − 1.25i)3-s + (2.16 − 3.74i)4-s + (−4.17 − 1.23i)6-s + (−0.445 + 0.257i)7-s − 5.83i·8-s + (−0.160 + 2.99i)9-s + (1.66 + 2.87i)11-s + (−7.27 + 1.74i)12-s + (−1.14 − 0.660i)13-s + (−0.646 + 1.11i)14-s + (−3.01 − 5.22i)16-s − 3.32i·17-s + (3.41 + 6.72i)18-s + 1.32·19-s + ⋯
L(s)  = 1  + (1.53 − 0.888i)2-s + (−0.687 − 0.725i)3-s + (1.08 − 1.87i)4-s + (−1.70 − 0.505i)6-s + (−0.168 + 0.0971i)7-s − 2.06i·8-s + (−0.0534 + 0.998i)9-s + (0.500 + 0.867i)11-s + (−2.10 + 0.503i)12-s + (−0.317 − 0.183i)13-s + (−0.172 + 0.299i)14-s + (−0.753 − 1.30i)16-s − 0.805i·17-s + (0.805 + 1.58i)18-s + 0.303·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.335 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.335 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.335 + 0.941i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.335 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26082 - 1.78835i\)
\(L(\frac12)\) \(\approx\) \(1.26082 - 1.78835i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.19 + 1.25i)T \)
5 \( 1 \)
good2 \( 1 + (-2.17 + 1.25i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (0.445 - 0.257i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.66 - 2.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.14 + 0.660i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.32iT - 17T^{2} \)
19 \( 1 - 1.32T + 19T^{2} \)
23 \( 1 + (-3.57 - 2.06i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.693 + 1.20i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.36 - 7.56i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.292iT - 37T^{2} \)
41 \( 1 + (-5.67 + 9.82i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (8.96 - 5.17i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.21 - 2.43i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 5.02iT - 53T^{2} \)
59 \( 1 + (2.51 - 4.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.67 + 6.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.18 + 4.72i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.99T + 71T^{2} \)
73 \( 1 + 6.05iT - 73T^{2} \)
79 \( 1 + (4.02 + 6.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.33 - 0.771i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (10.6 - 6.12i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.21130399375844023985631068053, −11.37613911828228982628276329329, −10.57090812023035406549977799460, −9.405618151769269473504748739224, −7.43222280134873561125155052703, −6.53717557791273232173072697564, −5.39452331535609976057513701970, −4.61986432348577365201740140442, −3.03522782412027395945825485058, −1.62811490229421640849011235136, 3.29854387106738523103019707870, 4.21092329718107548473621695374, 5.29038448716135308274966236181, 6.14343261077069396354874372579, 6.94836048155894673043956841204, 8.361617243942134208090698025255, 9.684203704831847340295808661105, 11.05979972449180139065850365845, 11.72106542427515137174378588974, 12.72513676907696277481559501930

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