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Properties

Label 2-30e2-20.7-c1-0-28
Degree $2$
Conductor $900$
Sign $-0.630 + 0.775i$
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  + (0.0912 − 1.41i)2-s + (−1.98 − 0.257i)4-s + (1.86 − 1.86i)7-s + (−0.544 + 2.77i)8-s + 0.728i·11-s + (3.12 − 3.12i)13-s + (−2.46 − 2.80i)14-s + (3.86 + 1.02i)16-s + (1.12 + 1.12i)17-s + 3.73·19-s + (1.02 + 0.0664i)22-s + (−5.83 − 5.83i)23-s + (−4.12 − 4.69i)26-s + (−4.18 + 3.22i)28-s + 2.64i·29-s + ⋯
L(s)  = 1  + (0.0645 − 0.997i)2-s + (−0.991 − 0.128i)4-s + (0.705 − 0.705i)7-s + (−0.192 + 0.981i)8-s + 0.219i·11-s + (0.866 − 0.866i)13-s + (−0.658 − 0.749i)14-s + (0.966 + 0.255i)16-s + (0.272 + 0.272i)17-s + 0.856·19-s + (0.219 + 0.0141i)22-s + (−1.21 − 1.21i)23-s + (−0.808 − 0.920i)26-s + (−0.790 + 0.608i)28-s + 0.490i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.630 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.630 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.630 + 0.775i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -0.630 + 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.649389 - 1.36505i\)
\(L(\frac12)\) \(\approx\) \(0.649389 - 1.36505i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0912 + 1.41i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-1.86 + 1.86i)T - 7iT^{2} \)
11 \( 1 - 0.728iT - 11T^{2} \)
13 \( 1 + (-3.12 + 3.12i)T - 13iT^{2} \)
17 \( 1 + (-1.12 - 1.12i)T + 17iT^{2} \)
19 \( 1 - 3.73T + 19T^{2} \)
23 \( 1 + (5.83 + 5.83i)T + 23iT^{2} \)
29 \( 1 - 2.64iT - 29T^{2} \)
31 \( 1 + 6.01iT - 31T^{2} \)
37 \( 1 + (3.12 + 3.12i)T + 37iT^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 + (5.10 + 5.10i)T + 43iT^{2} \)
47 \( 1 + (-2.09 + 2.09i)T - 47iT^{2} \)
53 \( 1 + (-0.484 + 0.484i)T - 53iT^{2} \)
59 \( 1 + 4.92T + 59T^{2} \)
61 \( 1 - 2.31T + 61T^{2} \)
67 \( 1 + (-5.10 + 5.10i)T - 67iT^{2} \)
71 \( 1 + 13.1iT - 71T^{2} \)
73 \( 1 + (3.96 - 3.96i)T - 73iT^{2} \)
79 \( 1 + 7.11T + 79T^{2} \)
83 \( 1 + (-3.55 - 3.55i)T + 83iT^{2} \)
89 \( 1 - 1.03iT - 89T^{2} \)
97 \( 1 + (-12.5 - 12.5i)T + 97iT^{2} \)
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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

−10.16362498585803973166751835264, −9.085231477389253999679052987318, −8.187728452726754815942955934293, −7.59770542747819315788189140428, −6.13009330122418530002101818622, −5.19073321394700387419946595816, −4.20288612528679100498303044975, −3.40488182811004410832130632834, −2.04441903337989233496878090304, −0.797338722993198515233733876793, 1.51151734728592028831414450572, 3.30092923139598956347071398899, 4.36992275868305774741173123142, 5.37697203694032111347849946711, 6.00060169103528016177791257832, 7.01456429584995192934994143230, 7.918863362942762123258545870473, 8.581607048872550030586544591517, 9.330409892996151581880536377006, 10.13687584049148305250368386981

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