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Properties

Label 2-300-4.3-c2-0-36
Degree $2$
Conductor $300$
Sign $-0.987 + 0.159i$
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  + (1.29 − 1.52i)2-s − 1.73i·3-s + (−0.637 − 3.94i)4-s + (−2.63 − 2.24i)6-s + 0.837i·7-s + (−6.83 − 4.14i)8-s − 2.99·9-s − 15.7i·11-s + (−6.83 + 1.10i)12-s + 5.18·13-s + (1.27 + 1.08i)14-s + (−15.1 + 5.03i)16-s − 27.3·17-s + (−3.88 + 4.56i)18-s − 17.9i·19-s + ⋯
L(s)  = 1  + (0.648 − 0.761i)2-s − 0.577i·3-s + (−0.159 − 0.987i)4-s + (−0.439 − 0.374i)6-s + 0.119i·7-s + (−0.854 − 0.518i)8-s − 0.333·9-s − 1.43i·11-s + (−0.569 + 0.0920i)12-s + 0.398·13-s + (0.0910 + 0.0775i)14-s + (−0.949 + 0.314i)16-s − 1.60·17-s + (−0.216 + 0.253i)18-s − 0.945i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.159i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.987 + 0.159i$
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: \((2,\ 300,\ (\ :1),\ -0.987 + 0.159i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.144369 - 1.80023i\)
\(L(\frac12)\) \(\approx\) \(0.144369 - 1.80023i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 + 1.52i)T \)
3 \( 1 + 1.73iT \)
5 \( 1 \)
good7 \( 1 - 0.837iT - 49T^{2} \)
11 \( 1 + 15.7iT - 121T^{2} \)
13 \( 1 - 5.18T + 169T^{2} \)
17 \( 1 + 27.3T + 289T^{2} \)
19 \( 1 + 17.9iT - 361T^{2} \)
23 \( 1 - 19.1iT - 529T^{2} \)
29 \( 1 - 45.6T + 841T^{2} \)
31 \( 1 - 13.6iT - 961T^{2} \)
37 \( 1 - 15.5T + 1.36e3T^{2} \)
41 \( 1 - 13.2T + 1.68e3T^{2} \)
43 \( 1 + 27.9iT - 1.84e3T^{2} \)
47 \( 1 + 55.6iT - 2.20e3T^{2} \)
53 \( 1 + 15.5T + 2.80e3T^{2} \)
59 \( 1 + 87.6iT - 3.48e3T^{2} \)
61 \( 1 - 38T + 3.72e3T^{2} \)
67 \( 1 - 92.2iT - 4.48e3T^{2} \)
71 \( 1 + 130. iT - 5.04e3T^{2} \)
73 \( 1 - 54.7T + 5.32e3T^{2} \)
79 \( 1 - 13.6iT - 6.24e3T^{2} \)
83 \( 1 + 59.0iT - 6.88e3T^{2} \)
89 \( 1 + 39.8T + 7.92e3T^{2} \)
97 \( 1 - 168.T + 9.40e3T^{2} \)
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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

−11.25976446241181477272769354067, −10.54759796918771937750469407977, −9.118739296234921463033485823068, −8.476310873471940502182199801743, −6.82795255682269746699286191923, −6.03994806002148817145043042779, −4.89376171146050178582818146254, −3.50878674978183564368675763170, −2.34414731753038604430487325307, −0.70807521980678229925563562651, 2.53048211116850676271119731807, 4.16063810637871996191045803163, 4.66960421430614309896562343949, 6.06551196871943439638916568141, 6.93193777672621239418885640230, 8.066349926938280263284396920129, 8.991036920927364893290370492853, 10.03193497525811215667985164643, 11.10532085456367327391593596005, 12.18919385661045745833431629770

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