Properties

Label 2-300-4.3-c2-0-5
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

\(L(\frac{3}{2})\) \(\approx\) \(0.0404866 - 0.722732i\)
\(L(\frac12)\) \(\approx\) \(0.0404866 - 0.722732i\)
\(L(2)\) 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

−11.61085217108177735751814324838, −10.97734602849784213710387235453, −9.883845676049147115156031588832, −9.004483633631729718210186223461, −8.486902083507835990114532599374, −7.20916590621286021636555373939, −6.02681951678452036736400738917, −5.31399808206552118680714492159, −3.88597423827901572785661086495, −1.93265448734993787335902132294, 0.43153919398526176434424569808, 1.88800396186055897657657732647, 3.39324805148440310685448830527, 4.56033601452852005560190214867, 6.49161344877504850789967673117, 7.20084006441738401872945863104, 8.440151561673237726134785241852, 8.952647627420599031114660904641, 10.39450929433013063244620668635, 10.94688728082115450968078848070

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