Properties

Label 2-300-20.19-c2-0-34
Degree $2$
Conductor $300$
Sign $-0.536 + 0.843i$
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  + (1.92 − 0.534i)2-s − 1.73·3-s + (3.42 − 2.05i)4-s + (−3.33 + 0.925i)6-s − 11.9·7-s + (5.51 − 5.79i)8-s + 2.99·9-s − 14.5i·11-s + (−5.94 + 3.56i)12-s − 22.4i·13-s + (−23.0 + 6.39i)14-s + (7.52 − 14.1i)16-s + 12.6i·17-s + (5.78 − 1.60i)18-s − 8.76i·19-s + ⋯
L(s)  = 1  + (0.963 − 0.267i)2-s − 0.577·3-s + (0.857 − 0.514i)4-s + (−0.556 + 0.154i)6-s − 1.71·7-s + (0.688 − 0.724i)8-s + 0.333·9-s − 1.32i·11-s + (−0.495 + 0.297i)12-s − 1.72i·13-s + (−1.64 + 0.456i)14-s + (0.470 − 0.882i)16-s + 0.746i·17-s + (0.321 − 0.0890i)18-s − 0.461i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.536 + 0.843i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.536 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.536 + 0.843i$
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: \((2,\ 300,\ (\ :1),\ -0.536 + 0.843i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.794072 - 1.44616i\)
\(L(\frac12)\) \(\approx\) \(0.794072 - 1.44616i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.92 + 0.534i)T \)
3 \( 1 + 1.73T \)
5 \( 1 \)
good7 \( 1 + 11.9T + 49T^{2} \)
11 \( 1 + 14.5iT - 121T^{2} \)
13 \( 1 + 22.4iT - 169T^{2} \)
17 \( 1 - 12.6iT - 289T^{2} \)
19 \( 1 + 8.76iT - 361T^{2} \)
23 \( 1 + 4.99T + 529T^{2} \)
29 \( 1 + 2.74T + 841T^{2} \)
31 \( 1 - 16.3iT - 961T^{2} \)
37 \( 1 - 32.4iT - 1.36e3T^{2} \)
41 \( 1 - 42.7T + 1.68e3T^{2} \)
43 \( 1 + 16.5T + 1.84e3T^{2} \)
47 \( 1 + 48.5T + 2.20e3T^{2} \)
53 \( 1 + 94.1iT - 2.80e3T^{2} \)
59 \( 1 - 43.2iT - 3.48e3T^{2} \)
61 \( 1 - 56.7T + 3.72e3T^{2} \)
67 \( 1 - 61.1T + 4.48e3T^{2} \)
71 \( 1 + 39.6iT - 5.04e3T^{2} \)
73 \( 1 - 99.5iT - 5.32e3T^{2} \)
79 \( 1 + 10.7iT - 6.24e3T^{2} \)
83 \( 1 - 140.T + 6.88e3T^{2} \)
89 \( 1 + 54.8T + 7.92e3T^{2} \)
97 \( 1 - 14.1iT - 9.40e3T^{2} \)
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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

−11.24905745892273548028202994894, −10.45622501533481944096654840954, −9.764445149587466273313763226963, −8.225256530598050937674663003626, −6.78595158133459669761681041624, −6.06137301876322604894683467606, −5.33311253231572629684573802129, −3.66300888205277502945341851697, −2.94903150571706995377093303137, −0.60404350698900908898287631610, 2.22725994310634659237363073724, 3.76050966814884468579735862096, 4.68305047043092736640597669054, 6.02327700915720692977877511259, 6.75104164898562337041168683209, 7.41358155331974812036952443620, 9.282109388644133528847255596995, 9.947410462502470454485013516786, 11.20866488936042089045424544603, 12.14850410494451471720252778273

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