Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 5^{2} $
Sign $0.766 - 0.641i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.192 − 1.40i)2-s − 3-s + (−1.92 − 0.540i)4-s + (−0.192 + 1.40i)6-s + 0.0802i·7-s + (−1.12 + 2.59i)8-s + 9-s + 2.41i·11-s + (1.92 + 0.540i)12-s − 5.26·13-s + (0.112 + 0.0154i)14-s + (3.41 + 2.08i)16-s − 0.255i·17-s + (0.192 − 1.40i)18-s + 6.95i·19-s + ⋯
L(s)  = 1  + (0.136 − 0.990i)2-s − 0.577·3-s + (−0.962 − 0.270i)4-s + (−0.0786 + 0.571i)6-s + 0.0303i·7-s + (−0.398 + 0.917i)8-s + 0.333·9-s + 0.728i·11-s + (0.555 + 0.155i)12-s − 1.46·13-s + (0.0300 + 0.00413i)14-s + (0.854 + 0.520i)16-s − 0.0620i·17-s + (0.0454 − 0.330i)18-s + 1.59i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $0.766 - 0.641i$
motivic weight  =  \(1\)
character  :  $\chi_{600} (349, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 600,\ (\ :1/2),\ 0.766 - 0.641i)\)
\(L(1)\)  \(\approx\)  \(0.634681 + 0.230579i\)
\(L(\frac12)\)  \(\approx\)  \(0.634681 + 0.230579i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-0.192 + 1.40i)T \)
3 \( 1 + T \)
5 \( 1 \)
good7 \( 1 - 0.0802iT - 7T^{2} \)
11 \( 1 - 2.41iT - 11T^{2} \)
13 \( 1 + 5.26T + 13T^{2} \)
17 \( 1 + 0.255iT - 17T^{2} \)
19 \( 1 - 6.95iT - 19T^{2} \)
23 \( 1 - 1.64iT - 23T^{2} \)
29 \( 1 - 4.51iT - 29T^{2} \)
31 \( 1 - 8.29T + 31T^{2} \)
37 \( 1 - 2.67T + 37T^{2} \)
41 \( 1 + 8.11T + 41T^{2} \)
43 \( 1 - 4.08T + 43T^{2} \)
47 \( 1 - 5.70iT - 47T^{2} \)
53 \( 1 + 11.5T + 53T^{2} \)
59 \( 1 - 12.6iT - 59T^{2} \)
61 \( 1 + 11.9iT - 61T^{2} \)
67 \( 1 + 7.27T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 - 12.0iT - 73T^{2} \)
79 \( 1 + 5.50T + 79T^{2} \)
83 \( 1 - 9.20T + 83T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 - 8.50iT - 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.69303216697405152618555414971, −10.01661547115051536292808071021, −9.476407909465143944811424501526, −8.202550574922483207906573673715, −7.25088598625747830100063122968, −5.96543818906426696969132675634, −4.99591608274929784726825121237, −4.22677373175138084900531837283, −2.84501346881526334164471579589, −1.54654888827323749865831004164, 0.39865249560478206022760976376, 2.85090310150836847616522388435, 4.41175275965834762692367783102, 5.05671091149576887048236805366, 6.11008374001671873279655624261, 6.89121901097284999559661634175, 7.72233659688665529442913425804, 8.705796652773342602233232319232, 9.602636136027772536539928122874, 10.41528573415327318381951766247

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