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Properties

Label 2-300-100.27-c1-0-16
Degree $2$
Conductor $300$
Sign $0.770 + 0.637i$
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  + (−1.14 + 0.827i)2-s + (0.891 − 0.453i)3-s + (0.631 − 1.89i)4-s + (−2.20 − 0.377i)5-s + (−0.646 + 1.25i)6-s + (0.312 − 0.312i)7-s + (0.844 + 2.69i)8-s + (0.587 − 0.809i)9-s + (2.84 − 1.39i)10-s + (−0.610 − 0.840i)11-s + (−0.298 − 1.97i)12-s + (5.63 − 0.892i)13-s + (−0.0999 + 0.616i)14-s + (−2.13 + 0.664i)15-s + (−3.20 − 2.39i)16-s + (3.24 − 6.36i)17-s + ⋯
L(s)  = 1  + (−0.811 + 0.584i)2-s + (0.514 − 0.262i)3-s + (0.315 − 0.948i)4-s + (−0.985 − 0.168i)5-s + (−0.263 + 0.513i)6-s + (0.118 − 0.118i)7-s + (0.298 + 0.954i)8-s + (0.195 − 0.269i)9-s + (0.898 − 0.439i)10-s + (−0.184 − 0.253i)11-s + (−0.0862 − 0.570i)12-s + (1.56 − 0.247i)13-s + (−0.0267 + 0.164i)14-s + (−0.551 + 0.171i)15-s + (−0.800 − 0.599i)16-s + (0.786 − 1.54i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.770 + 0.637i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.770 + 0.637i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.837318 - 0.301753i\)
\(L(\frac12)\) \(\approx\) \(0.837318 - 0.301753i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.14 - 0.827i)T \)
3 \( 1 + (-0.891 + 0.453i)T \)
5 \( 1 + (2.20 + 0.377i)T \)
good7 \( 1 + (-0.312 + 0.312i)T - 7iT^{2} \)
11 \( 1 + (0.610 + 0.840i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-5.63 + 0.892i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (-3.24 + 6.36i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (2.11 + 6.50i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (2.01 + 0.318i)T + (21.8 + 7.10i)T^{2} \)
29 \( 1 + (-0.380 - 0.123i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.67 - 1.19i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.746 + 4.71i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (-0.813 - 0.590i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-5.34 - 5.34i)T + 43iT^{2} \)
47 \( 1 + (-3.64 - 7.15i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (-4.32 - 8.49i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (9.01 + 6.54i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.00534 + 0.00388i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (11.0 + 5.61i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (-3.28 - 1.06i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.742 - 4.68i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (4.96 - 15.2i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.00 + 5.89i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (1.83 + 2.53i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-1.36 + 0.695i)T + (57.0 - 78.4i)T^{2} \)
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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

−11.27374538931785797429893402487, −10.84004186918788568592007494486, −9.336045777564646777858934091079, −8.728969881605856299070410248963, −7.78964476931208112983869441719, −7.16970001224769413286786569085, −5.92258524116702655671200914122, −4.53378846130591469725990652835, −2.96561169217227801271227144429, −0.886716408207889199510855476207, 1.72459978886284598889037216753, 3.53917087953481988147640842842, 3.99245661807823000305590472579, 6.09745541570196782983931399377, 7.48778427962878366835021145494, 8.302507740480519249834817616845, 8.757282034933428870953716007859, 10.23642420960530188287301146994, 10.64546255807731906914860149856, 11.75462106678606293449372115460

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