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Properties

Label 2-300-75.8-c1-0-8
Degree $2$
Conductor $300$
Sign $0.329 + 0.944i$
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  + (0.767 − 1.55i)3-s + (1.73 − 1.40i)5-s + (−0.636 + 0.636i)7-s + (−1.82 − 2.38i)9-s + (2.06 − 0.672i)11-s + (0.777 − 0.396i)13-s + (−0.850 − 3.77i)15-s + (0.108 + 0.682i)17-s + (−3.00 + 4.14i)19-s + (0.499 + 1.47i)21-s + (−4.93 − 2.51i)23-s + (1.04 − 4.89i)25-s + (−5.09 + 1.00i)27-s + (4.84 − 3.52i)29-s + (6.26 + 4.54i)31-s + ⋯
L(s)  = 1  + (0.443 − 0.896i)3-s + (0.777 − 0.629i)5-s + (−0.240 + 0.240i)7-s + (−0.607 − 0.794i)9-s + (0.623 − 0.202i)11-s + (0.215 − 0.109i)13-s + (−0.219 − 0.975i)15-s + (0.0262 + 0.165i)17-s + (−0.690 + 0.949i)19-s + (0.109 + 0.322i)21-s + (−1.02 − 0.523i)23-s + (0.208 − 0.978i)25-s + (−0.981 + 0.192i)27-s + (0.900 − 0.654i)29-s + (1.12 + 0.817i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.329 + 0.944i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.329 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31290 - 0.931929i\)
\(L(\frac12)\) \(\approx\) \(1.31290 - 0.931929i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.767 + 1.55i)T \)
5 \( 1 + (-1.73 + 1.40i)T \)
good7 \( 1 + (0.636 - 0.636i)T - 7iT^{2} \)
11 \( 1 + (-2.06 + 0.672i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.777 + 0.396i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (-0.108 - 0.682i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (3.00 - 4.14i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (4.93 + 2.51i)T + (13.5 + 18.6i)T^{2} \)
29 \( 1 + (-4.84 + 3.52i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-6.26 - 4.54i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.550 + 1.08i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (-2.90 - 0.945i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-8.01 - 8.01i)T + 43iT^{2} \)
47 \( 1 + (7.69 + 1.21i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (0.429 - 2.71i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (3.57 - 10.9i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.58 - 7.95i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (9.44 - 1.49i)T + (63.7 - 20.7i)T^{2} \)
71 \( 1 + (3.16 + 4.35i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-7.59 + 14.9i)T + (-42.9 - 59.0i)T^{2} \)
79 \( 1 + (-2.00 - 2.76i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (5.95 - 0.942i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 + (-4.99 - 15.3i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (0.0903 - 0.570i)T + (-92.2 - 29.9i)T^{2} \)
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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

−12.02485334749015961637583727142, −10.50921519757994132751822512629, −9.496567242689445896662048142320, −8.631261408520432083488831853323, −7.920024788076706161932857978893, −6.35069200004137192133500964843, −6.01776810909453268608537701751, −4.29634393260827853006658219295, −2.68644806535747257536821530304, −1.34176910984256797568189381020, 2.27504243936063740163897142280, 3.53135226217719278577809498773, 4.69478892077630250522514194326, 6.00448345452297119659072287974, 6.96275110880674693141242703129, 8.320201572248662330406164309960, 9.338376891735645672777755159802, 9.972693585784643438720347459871, 10.78987359276853970443145011710, 11.67348038565106364023458946255

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