Properties

Label 2-300-4.3-c2-0-10
Degree $2$
Conductor $300$
Sign $-0.993 - 0.111i$
Analytic cond. $8.17440$
Root an. cond. $2.85909$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.33 + 1.49i)2-s + 1.73i·3-s + (−0.446 + 3.97i)4-s + (−2.58 + 2.30i)6-s + 6.56i·7-s + (−6.52 + 4.63i)8-s − 2.99·9-s − 2.26i·11-s + (−6.88 − 0.773i)12-s − 14.8·13-s + (−9.79 + 8.75i)14-s + (−15.6 − 3.55i)16-s + 26.8·17-s + (−3.99 − 4.47i)18-s − 10.8i·19-s + ⋯
L(s)  = 1  + (0.666 + 0.745i)2-s + 0.577i·3-s + (−0.111 + 0.993i)4-s + (−0.430 + 0.384i)6-s + 0.938i·7-s + (−0.815 + 0.579i)8-s − 0.333·9-s − 0.206i·11-s + (−0.573 − 0.0644i)12-s − 1.14·13-s + (−0.699 + 0.625i)14-s + (−0.975 − 0.221i)16-s + 1.57·17-s + (−0.222 − 0.248i)18-s − 0.572i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.111i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.993 - 0.111i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ -0.993 - 0.111i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.103229 + 1.84276i\)
\(L(\frac12)\) \(\approx\) \(0.103229 + 1.84276i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 - 1.49i)T \)
3 \( 1 - 1.73iT \)
5 \( 1 \)
good7 \( 1 - 6.56iT - 49T^{2} \)
11 \( 1 + 2.26iT - 121T^{2} \)
13 \( 1 + 14.8T + 169T^{2} \)
17 \( 1 - 26.8T + 289T^{2} \)
19 \( 1 + 10.8iT - 361T^{2} \)
23 \( 1 - 36.4iT - 529T^{2} \)
29 \( 1 + 35.2T + 841T^{2} \)
31 \( 1 - 23.8iT - 961T^{2} \)
37 \( 1 - 54.7T + 1.36e3T^{2} \)
41 \( 1 + 23.8T + 1.68e3T^{2} \)
43 \( 1 + 56.2iT - 1.84e3T^{2} \)
47 \( 1 - 51.4iT - 2.20e3T^{2} \)
53 \( 1 - 30.6T + 2.80e3T^{2} \)
59 \( 1 - 6.92iT - 3.48e3T^{2} \)
61 \( 1 - 107.T + 3.72e3T^{2} \)
67 \( 1 - 111. iT - 4.48e3T^{2} \)
71 \( 1 - 31.3iT - 5.04e3T^{2} \)
73 \( 1 - 110.T + 5.32e3T^{2} \)
79 \( 1 + 59.0iT - 6.24e3T^{2} \)
83 \( 1 - 142. iT - 6.88e3T^{2} \)
89 \( 1 - 7.14T + 7.92e3T^{2} \)
97 \( 1 + 126.T + 9.40e3T^{2} \)
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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

−12.03931232457629312896189886308, −11.36676619332213688099862943396, −9.840489800728361390071592892877, −9.120836138670063475986924481534, −8.008282136314744816988757754047, −7.11145065720929745604642026065, −5.60419151803816776309592930940, −5.27349133984326717755971188459, −3.80110532091597969833679552851, −2.66902590430904938673351027541, 0.72229386621204027289868711437, 2.25619462362142325213700437744, 3.62752075446351502791654068577, 4.79843528605876422811484743095, 5.95297816634678627266246666874, 7.09630612036242547097683791613, 8.015465614729292124380973758361, 9.624827312318583672584937704048, 10.19861291217697845843411517524, 11.23040616825151254286852566143

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