Properties

Label 2-300-25.6-c1-0-3
Degree $2$
Conductor $300$
Sign $0.837 + 0.546i$
Analytic cond. $2.39551$
Root an. cond. $1.54774$
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  + (0.809 − 0.587i)3-s + (1.99 − 1.00i)5-s − 0.0883·7-s + (0.309 − 0.951i)9-s + (0.701 + 2.15i)11-s + (0.819 − 2.52i)13-s + (1.02 − 1.98i)15-s + (−1.68 − 1.22i)17-s + (−1.42 − 1.03i)19-s + (−0.0714 + 0.0519i)21-s + (1.46 + 4.50i)23-s + (2.98 − 4.00i)25-s + (−0.309 − 0.951i)27-s + (−2.99 + 2.17i)29-s + (3.32 + 2.41i)31-s + ⋯
L(s)  = 1  + (0.467 − 0.339i)3-s + (0.893 − 0.448i)5-s − 0.0333·7-s + (0.103 − 0.317i)9-s + (0.211 + 0.650i)11-s + (0.227 − 0.699i)13-s + (0.265 − 0.512i)15-s + (−0.409 − 0.297i)17-s + (−0.326 − 0.237i)19-s + (−0.0155 + 0.0113i)21-s + (0.305 + 0.940i)23-s + (0.597 − 0.801i)25-s + (−0.0594 − 0.183i)27-s + (−0.556 + 0.404i)29-s + (0.596 + 0.433i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.837 + 0.546i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.837 + 0.546i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62459 - 0.482761i\)
\(L(\frac12)\) \(\approx\) \(1.62459 - 0.482761i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-1.99 + 1.00i)T \)
good7 \( 1 + 0.0883T + 7T^{2} \)
11 \( 1 + (-0.701 - 2.15i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.819 + 2.52i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.68 + 1.22i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.42 + 1.03i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.46 - 4.50i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (2.99 - 2.17i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-3.32 - 2.41i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.19 + 6.77i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.03 - 6.26i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.79T + 43T^{2} \)
47 \( 1 + (8.17 - 5.94i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.777 + 0.565i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.77 - 8.53i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.92 - 9.00i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (11.2 + 8.17i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (4.97 - 3.61i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.975 + 3.00i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (10.8 - 7.84i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-12.7 - 9.29i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (3.16 + 9.74i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-10.3 + 7.52i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.77432616850038955607264082644, −10.59617637267028517456033248324, −9.596969116724887425795775344108, −8.953232727197257464488298305437, −7.85467783586010133907018045986, −6.78479324261527201986077635806, −5.71318989296061793644711946048, −4.54513802939548934376194359505, −2.92355213689828465024673889288, −1.53963131806076570823056915314, 1.99150017254603040029405385034, 3.30966819492146361550953339878, 4.62882013492822122160086438804, 6.02155833461882433970490822314, 6.77870076065054155968775090938, 8.225155488055113601146561016220, 9.052798021606660853530913945967, 9.933552620499364648618343679506, 10.76933683734906332987525722836, 11.64880004008205842035894678374

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