Properties

Label 2-300-20.19-c2-0-21
Degree $2$
Conductor $300$
Sign $-0.591 + 0.806i$
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  + (0.169 − 1.99i)2-s − 1.73·3-s + (−3.94 − 0.675i)4-s + (−0.293 + 3.45i)6-s + 12.3·7-s + (−2.01 + 7.74i)8-s + 2.99·9-s − 11.0i·11-s + (6.82 + 1.16i)12-s − 2.82i·13-s + (2.10 − 24.7i)14-s + (15.0 + 5.32i)16-s + 6.52i·17-s + (0.508 − 5.97i)18-s − 27.9i·19-s + ⋯
L(s)  = 1  + (0.0847 − 0.996i)2-s − 0.577·3-s + (−0.985 − 0.168i)4-s + (−0.0489 + 0.575i)6-s + 1.77·7-s + (−0.251 + 0.967i)8-s + 0.333·9-s − 1.00i·11-s + (0.569 + 0.0974i)12-s − 0.216i·13-s + (0.150 − 1.76i)14-s + (0.942 + 0.332i)16-s + 0.383i·17-s + (0.0282 − 0.332i)18-s − 1.47i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 + 0.806i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.591 + 0.806i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ -0.591 + 0.806i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.626178 - 1.23654i\)
\(L(\frac12)\) \(\approx\) \(0.626178 - 1.23654i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.169 + 1.99i)T \)
3 \( 1 + 1.73T \)
5 \( 1 \)
good7 \( 1 - 12.3T + 49T^{2} \)
11 \( 1 + 11.0iT - 121T^{2} \)
13 \( 1 + 2.82iT - 169T^{2} \)
17 \( 1 - 6.52iT - 289T^{2} \)
19 \( 1 + 27.9iT - 361T^{2} \)
23 \( 1 - 7.90T + 529T^{2} \)
29 \( 1 + 50.7T + 841T^{2} \)
31 \( 1 + 36.3iT - 961T^{2} \)
37 \( 1 + 18.9iT - 1.36e3T^{2} \)
41 \( 1 - 5.30T + 1.68e3T^{2} \)
43 \( 1 - 45.5T + 1.84e3T^{2} \)
47 \( 1 - 11.7T + 2.20e3T^{2} \)
53 \( 1 + 41.1iT - 2.80e3T^{2} \)
59 \( 1 - 10.7iT - 3.48e3T^{2} \)
61 \( 1 - 56.1T + 3.72e3T^{2} \)
67 \( 1 + 16.1T + 4.48e3T^{2} \)
71 \( 1 + 66.1iT - 5.04e3T^{2} \)
73 \( 1 + 15.6iT - 5.32e3T^{2} \)
79 \( 1 - 123. iT - 6.24e3T^{2} \)
83 \( 1 - 99.6T + 6.88e3T^{2} \)
89 \( 1 + 101.T + 7.92e3T^{2} \)
97 \( 1 - 127. iT - 9.40e3T^{2} \)
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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

−11.02369505742123600392842620295, −10.92762469838908193467538632281, −9.395080273655115243729291972493, −8.497238948182549365056931517294, −7.54432469770334698440757552082, −5.76463275806659948465636421671, −4.99579307528422239074732499027, −3.93133387534497842607880263694, −2.22160864579171495833824578182, −0.797377112131599402338939739939, 1.55916413286521470525823595412, 4.10289474983015202211536535562, 4.96737637767451009430120086997, 5.75961716805033972305128143721, 7.15828093925966569140246887645, 7.75089271389876864358946755068, 8.782555266835479178432428812500, 9.899707315171950341859908164584, 10.94404891645348638231869146573, 11.95467109397428469919659262172

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