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Properties

Label 2-30e2-100.23-c1-0-28
Degree $2$
Conductor $900$
Sign $0.860 + 0.509i$
Analytic cond. $7.18653$
Root an. cond. $2.68077$
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  + (−1.37 − 0.343i)2-s + (1.76 + 0.941i)4-s + (1.18 − 1.89i)5-s + (−1.89 − 1.89i)7-s + (−2.09 − 1.89i)8-s + (−2.27 + 2.20i)10-s + (2.99 + 4.12i)11-s + (0.433 + 2.73i)13-s + (1.94 + 3.24i)14-s + (2.22 + 3.32i)16-s + (4.16 + 2.12i)17-s + (1.55 + 4.77i)19-s + (3.87 − 2.23i)20-s + (−2.69 − 6.69i)22-s + (0.699 − 4.41i)23-s + ⋯
L(s)  = 1  + (−0.970 − 0.242i)2-s + (0.882 + 0.470i)4-s + (0.528 − 0.849i)5-s + (−0.715 − 0.715i)7-s + (−0.741 − 0.670i)8-s + (−0.718 + 0.695i)10-s + (0.904 + 1.24i)11-s + (0.120 + 0.758i)13-s + (0.520 + 0.868i)14-s + (0.556 + 0.830i)16-s + (1.01 + 0.514i)17-s + (0.355 + 1.09i)19-s + (0.865 − 0.500i)20-s + (−0.575 − 1.42i)22-s + (0.145 − 0.921i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.860 + 0.509i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.860 + 0.509i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09353 - 0.299355i\)
\(L(\frac12)\) \(\approx\) \(1.09353 - 0.299355i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 + 0.343i)T \)
3 \( 1 \)
5 \( 1 + (-1.18 + 1.89i)T \)
good7 \( 1 + (1.89 + 1.89i)T + 7iT^{2} \)
11 \( 1 + (-2.99 - 4.12i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.433 - 2.73i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-4.16 - 2.12i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-1.55 - 4.77i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.699 + 4.41i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-0.211 - 0.0688i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-7.01 + 2.28i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.32 + 0.684i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (2.54 + 1.84i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (2.68 - 2.68i)T - 43iT^{2} \)
47 \( 1 + (-7.84 + 3.99i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (1.64 - 0.837i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (7.33 + 5.32i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.16 + 0.845i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.53 + 3.01i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (-5.45 - 1.77i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.0187 + 0.00296i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-2.08 + 6.41i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.99 + 1.52i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (-0.509 - 0.700i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (5.07 + 9.96i)T + (-57.0 + 78.4i)T^{2} \)
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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

−9.883098560161699612874885323778, −9.442458136473925072406516247195, −8.503642209032845090471682949163, −7.63365470527005185574266461955, −6.68527131222564184326436701524, −6.04853365943968150928450013059, −4.50420007077532872798980789079, −3.60962914362736533268987048727, −2.02718297658419592475644634990, −1.03250668309796975640215138884, 1.03232038424275679166414337885, 2.77914799091030366394698981485, 3.22810604659451047839474352279, 5.43850620638225096367628014928, 6.04068788065592771127177021991, 6.74587807492378136996716960266, 7.66124119859017912515303612027, 8.666417763959280745488744057206, 9.415359415797374270344090617941, 9.929257634694871155915619161859

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