Properties

Label 2-300-300.71-c1-0-6
Degree $2$
Conductor $300$
Sign $0.850 - 0.525i$
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  + (−0.936 − 1.05i)2-s + (1.00 − 1.40i)3-s + (−0.246 + 1.98i)4-s + (−1.49 + 1.66i)5-s + (−2.43 + 0.250i)6-s + 4.44i·7-s + (2.33 − 1.59i)8-s + (−0.967 − 2.83i)9-s + (3.16 + 0.0306i)10-s + (−1.38 + 4.26i)11-s + (2.54 + 2.34i)12-s + (1.09 + 3.38i)13-s + (4.71 − 4.16i)14-s + (0.831 + 3.78i)15-s + (−3.87 − 0.979i)16-s + (0.229 + 0.316i)17-s + ⋯
L(s)  = 1  + (−0.662 − 0.749i)2-s + (0.582 − 0.813i)3-s + (−0.123 + 0.992i)4-s + (−0.669 + 0.743i)5-s + (−0.994 + 0.102i)6-s + 1.68i·7-s + (0.825 − 0.564i)8-s + (−0.322 − 0.946i)9-s + (0.999 + 0.00969i)10-s + (−0.417 + 1.28i)11-s + (0.735 + 0.677i)12-s + (0.304 + 0.938i)13-s + (1.26 − 1.11i)14-s + (0.214 + 0.976i)15-s + (−0.969 − 0.244i)16-s + (0.0557 + 0.0767i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.850 - 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.812441 + 0.230593i\)
\(L(\frac12)\) \(\approx\) \(0.812441 + 0.230593i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.936 + 1.05i)T \)
3 \( 1 + (-1.00 + 1.40i)T \)
5 \( 1 + (1.49 - 1.66i)T \)
good7 \( 1 - 4.44iT - 7T^{2} \)
11 \( 1 + (1.38 - 4.26i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.09 - 3.38i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.229 - 0.316i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.18 + 1.63i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.927 - 2.85i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-4.65 + 6.40i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.713 + 0.981i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.145 + 0.447i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-7.88 + 2.56i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 - 5.44iT - 43T^{2} \)
47 \( 1 + (-5.11 - 3.71i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.31 - 1.81i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.692 - 2.13i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.80 - 5.55i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-4.83 - 6.65i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-1.00 - 0.727i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.53 + 13.9i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.87 + 8.07i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.74 - 3.44i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (13.5 + 4.39i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (-9.45 - 6.86i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.00068307222339420135362525560, −11.11020328702645975809133441494, −9.731261464712790392037240843860, −9.017513387561357508761718772427, −8.088262429389003995232036435547, −7.31344988891266466884958537516, −6.25057214878836505749436700228, −4.25658522889321665716483245074, −2.77231612614275655116190037539, −2.09052818955170425518468181477, 0.73562031364516853250992230084, 3.48003144440230287502900153537, 4.52778160656108155354499240208, 5.58641248861956376393369085159, 7.13949359858951308946954071148, 8.197115399919434444499125129379, 8.425420940234819807147551329861, 9.712738114717724591985041057610, 10.71548513283249240169110031193, 10.94303659093944855898176761468

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