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Properties

Label 2-300-75.17-c1-0-3
Degree $2$
Conductor $300$
Sign $0.646 - 0.763i$
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  + (−1.13 + 1.31i)3-s + (2.14 − 0.633i)5-s + (0.907 + 0.907i)7-s + (−0.440 − 2.96i)9-s + (1.03 − 0.337i)11-s + (2.26 + 4.44i)13-s + (−1.59 + 3.52i)15-s + (2.19 − 0.347i)17-s + (−2.39 + 3.30i)19-s + (−2.21 + 0.163i)21-s + (−2.13 + 4.18i)23-s + (4.19 − 2.71i)25-s + (4.39 + 2.77i)27-s + (0.981 − 0.712i)29-s + (0.992 + 0.720i)31-s + ⋯
L(s)  = 1  + (−0.653 + 0.757i)3-s + (0.959 − 0.283i)5-s + (0.343 + 0.343i)7-s + (−0.146 − 0.989i)9-s + (0.313 − 0.101i)11-s + (0.628 + 1.23i)13-s + (−0.411 + 0.911i)15-s + (0.532 − 0.0843i)17-s + (−0.550 + 0.757i)19-s + (−0.484 + 0.0357i)21-s + (−0.445 + 0.873i)23-s + (0.839 − 0.543i)25-s + (0.844 + 0.534i)27-s + (0.182 − 0.132i)29-s + (0.178 + 0.129i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 - 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.646 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.646 - 0.763i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.646 - 0.763i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16762 + 0.541287i\)
\(L(\frac12)\) \(\approx\) \(1.16762 + 0.541287i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.13 - 1.31i)T \)
5 \( 1 + (-2.14 + 0.633i)T \)
good7 \( 1 + (-0.907 - 0.907i)T + 7iT^{2} \)
11 \( 1 + (-1.03 + 0.337i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (-2.26 - 4.44i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-2.19 + 0.347i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (2.39 - 3.30i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (2.13 - 4.18i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (-0.981 + 0.712i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.992 - 0.720i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-6.47 + 3.29i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (8.57 + 2.78i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-1.48 + 1.48i)T - 43iT^{2} \)
47 \( 1 + (-0.0645 + 0.407i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (13.7 + 2.18i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (-3.76 + 11.6i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.344 - 1.05i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (1.22 + 7.74i)T + (-63.7 + 20.7i)T^{2} \)
71 \( 1 + (6.04 + 8.32i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-5.93 - 3.02i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (5.56 + 7.66i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.79 - 17.6i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (-4.10 - 12.6i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (15.3 + 2.42i)T + (92.2 + 29.9i)T^{2} \)
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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

−11.75571523869275090713246423856, −10.95910708650898901189793257044, −9.895792918742870280317061436590, −9.295788971018529115407719491996, −8.326641696620081731578426102739, −6.59364443120937429837373448903, −5.86826168814586652051954307684, −4.89025669417410184257846358211, −3.71025141617754350365211730383, −1.71639216613743298259961495148, 1.23968184451913104005894868358, 2.76702278573533390354109192344, 4.70891162078864930309436754144, 5.85913996620450766621195957567, 6.52315413289014326013600807406, 7.66011101560686888163020297834, 8.617604121587083190979571806805, 10.05521105586436605776351796411, 10.68157946507399839511911660112, 11.54518466308028953090019341189

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