Properties

Label 2-300-20.19-c2-0-17
Degree $2$
Conductor $300$
Sign $0.997 - 0.0769i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0769i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.997 - 0.0769i$
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: \((2,\ 300,\ (\ :1),\ 0.997 - 0.0769i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.56739 + 0.0603677i\)
\(L(\frac12)\) \(\approx\) \(1.56739 + 0.0603677i\)
\(L(2)\) 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

−11.26989329534391553505232546416, −10.72883881531666937666654096315, −9.133704690110111997234554454723, −8.866082252002472469953954529138, −7.82096683463357518172823616852, −7.04087904109109930534860613909, −5.68058539144558019273891752661, −4.38984856074442004550955772406, −2.51982794840729700792226930622, −1.25532565454947277615486933516, 1.37197105480943763882842600594, 2.49620246120529837286403489245, 4.10274392264708794607548257577, 5.49249770560154358284287965208, 7.16529788961955212851640649683, 8.006621536580683919091589751897, 8.382854527137161843355214253304, 9.716853277317340472186795129484, 10.44802006674984044548459559028, 11.27383065134341913572508097424

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