Properties

Label 2-300-300.179-c1-0-37
Degree $2$
Conductor $300$
Sign $0.999 - 0.0190i$
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  + (1.17 − 0.779i)2-s + (1.19 + 1.25i)3-s + (0.784 − 1.83i)4-s + (0.0402 + 2.23i)5-s + (2.38 + 0.543i)6-s − 1.15·7-s + (−0.509 − 2.78i)8-s + (−0.134 + 2.99i)9-s + (1.79 + 2.60i)10-s + (0.602 − 1.85i)11-s + (3.24 − 1.22i)12-s + (6.39 − 2.07i)13-s + (−1.36 + 0.898i)14-s + (−2.75 + 2.72i)15-s + (−2.76 − 2.88i)16-s + (−4.07 + 2.95i)17-s + ⋯
L(s)  = 1  + (0.834 − 0.551i)2-s + (0.691 + 0.722i)3-s + (0.392 − 0.919i)4-s + (0.0179 + 0.999i)5-s + (0.975 + 0.222i)6-s − 0.435·7-s + (−0.179 − 0.983i)8-s + (−0.0447 + 0.998i)9-s + (0.566 + 0.824i)10-s + (0.181 − 0.559i)11-s + (0.935 − 0.352i)12-s + (1.77 − 0.575i)13-s + (−0.363 + 0.240i)14-s + (−0.710 + 0.703i)15-s + (−0.692 − 0.721i)16-s + (−0.987 + 0.717i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0190i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.999 - 0.0190i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.999 - 0.0190i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.40334 + 0.0228481i\)
\(L(\frac12)\) \(\approx\) \(2.40334 + 0.0228481i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.17 + 0.779i)T \)
3 \( 1 + (-1.19 - 1.25i)T \)
5 \( 1 + (-0.0402 - 2.23i)T \)
good7 \( 1 + 1.15T + 7T^{2} \)
11 \( 1 + (-0.602 + 1.85i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-6.39 + 2.07i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (4.07 - 2.95i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.112 - 0.155i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (5.77 + 1.87i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-0.480 + 0.661i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.70 + 5.10i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.47 + 0.480i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-6.34 + 2.06i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + 4.89T + 43T^{2} \)
47 \( 1 + (-2.70 + 3.72i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-4.10 - 2.98i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.85 + 8.80i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.91 - 5.88i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (7.13 - 5.18i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-10.7 - 7.80i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.62 - 1.50i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-6.10 + 8.39i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.95 - 10.9i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (1.79 + 0.583i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (5.18 - 7.13i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.44314793820254490542924903422, −10.81940068088660420755429522269, −10.22336062006368784298231212588, −9.130612306646739940384596693148, −8.017564981599037045608742600558, −6.46111127135453840919482446121, −5.78779326684695544311692722026, −4.01535790518923226802636362603, −3.50797171684180098565182907518, −2.26600303995188692842663678077, 1.84151319320130544346019272683, 3.50741189348113458183533764354, 4.48839780010613380638735519725, 5.96839501823982727356605206710, 6.72783777506003304762790410723, 7.83781196809983907015626900119, 8.730651416383993448423970483006, 9.373277599036469156493105067736, 11.24343947077638318094840602874, 12.14100074914655311675679608903

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