Properties

Label 2-300-25.9-c1-0-4
Degree $2$
Conductor $300$
Sign $0.179 + 0.983i$
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  + (0.951 − 0.309i)3-s + (−1.98 − 1.02i)5-s − 3.54i·7-s + (0.809 − 0.587i)9-s + (1.78 + 1.29i)11-s + (−4.21 − 5.80i)13-s + (−2.20 − 0.358i)15-s + (6.05 + 1.96i)17-s + (0.715 − 2.20i)19-s + (−1.09 − 3.37i)21-s + (−1.27 + 1.76i)23-s + (2.90 + 4.06i)25-s + (0.587 − 0.809i)27-s + (−0.262 − 0.806i)29-s + (−1.32 + 4.09i)31-s + ⋯
L(s)  = 1  + (0.549 − 0.178i)3-s + (−0.889 − 0.457i)5-s − 1.34i·7-s + (0.269 − 0.195i)9-s + (0.538 + 0.390i)11-s + (−1.17 − 1.61i)13-s + (−0.569 − 0.0925i)15-s + (1.46 + 0.477i)17-s + (0.164 − 0.504i)19-s + (−0.239 − 0.736i)21-s + (−0.266 + 0.367i)23-s + (0.581 + 0.813i)25-s + (0.113 − 0.155i)27-s + (−0.0486 − 0.149i)29-s + (−0.238 + 0.734i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.179 + 0.983i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.179 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.984983 - 0.821503i\)
\(L(\frac12)\) \(\approx\) \(0.984983 - 0.821503i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 + (1.98 + 1.02i)T \)
good7 \( 1 + 3.54iT - 7T^{2} \)
11 \( 1 + (-1.78 - 1.29i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (4.21 + 5.80i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-6.05 - 1.96i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.715 + 2.20i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (1.27 - 1.76i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.262 + 0.806i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.32 - 4.09i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.24 - 5.84i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.08 + 0.790i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.18iT - 43T^{2} \)
47 \( 1 + (5.75 - 1.87i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-11.3 + 3.69i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (10.0 - 7.33i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-5.59 - 4.06i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-4.50 - 1.46i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-4.25 - 13.1i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.640 - 0.881i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.80 - 5.55i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (11.9 + 3.87i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (5.68 + 4.12i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-17.2 + 5.60i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.70296713961879175729411434229, −10.38140774577772311990165233250, −9.831324334320860003782044471049, −8.442685726087277766317344128250, −7.60217866388042628815678030148, −7.14614905013750159404745765907, −5.32006591786436844908980156520, −4.13428885632573439943912224912, −3.18037495476153487062494803698, −0.961941630125111895059538128162, 2.29631494461843530007588277442, 3.48789650025403627787696165306, 4.69844002929040538105253177368, 6.07212077052436709889662788141, 7.27857962042715345875066838348, 8.137079052661540020638030259476, 9.192958796325453151250011197669, 9.792275616378617615762385604476, 11.28538893746646122523295418274, 11.94501757037074332162832078784

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