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Properties

Label 2-600-40.29-c1-0-10
Degree $2$
Conductor $600$
Sign $0.987 - 0.155i$
Analytic cond. $4.79102$
Root an. cond. $2.18884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.144 − 1.40i)2-s + 3-s + (−1.95 − 0.406i)4-s + (0.144 − 1.40i)6-s + 3.62i·7-s + (−0.855 + 2.69i)8-s + 9-s + 6.20i·11-s + (−1.95 − 0.406i)12-s + 0.578·13-s + (5.10 + 0.524i)14-s + (3.66 + 1.59i)16-s − 1.42i·17-s + (0.144 − 1.40i)18-s + 5.62i·19-s + ⋯
L(s)  = 1  + (0.102 − 0.994i)2-s + 0.577·3-s + (−0.979 − 0.203i)4-s + (0.0590 − 0.574i)6-s + 1.37i·7-s + (−0.302 + 0.953i)8-s + 0.333·9-s + 1.87i·11-s + (−0.565 − 0.117i)12-s + 0.160·13-s + (1.36 + 0.140i)14-s + (0.917 + 0.398i)16-s − 0.344i·17-s + (0.0340 − 0.331i)18-s + 1.29i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.987 - 0.155i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.987 - 0.155i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55233 + 0.121641i\)
\(L(\frac12)\) \(\approx\) \(1.55233 + 0.121641i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.144 + 1.40i)T \)
3 \( 1 - T \)
5 \( 1 \)
good7 \( 1 - 3.62iT - 7T^{2} \)
11 \( 1 - 6.20iT - 11T^{2} \)
13 \( 1 - 0.578T + 13T^{2} \)
17 \( 1 + 1.42iT - 17T^{2} \)
19 \( 1 - 5.62iT - 19T^{2} \)
23 \( 1 + 5.62iT - 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 + 2.57T + 31T^{2} \)
37 \( 1 + 7.83T + 37T^{2} \)
41 \( 1 - 5.25T + 41T^{2} \)
43 \( 1 - 7.25T + 43T^{2} \)
47 \( 1 + 6.78iT - 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 2.20iT - 59T^{2} \)
61 \( 1 - 12.4iT - 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 8.41T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 5.42T + 79T^{2} \)
83 \( 1 + 3.25T + 83T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + 4.84iT - 97T^{2} \)
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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

−10.51482833635814917453412398865, −9.847291620308096630195647339844, −9.085020610116081631572863739448, −8.404299869066912667458707923381, −7.31608498815505600364533115793, −5.89623351596804595709932921843, −4.88130287791933523267089856338, −3.91080918901735059205203971493, −2.55400430444505181065966949595, −1.88669363849307231964633850692, 0.833535042035113445271247040078, 3.28471518732951362324936581607, 3.94944728548798118360630720725, 5.18402161556748117293998851341, 6.25997649880840775238660231351, 7.17774130201074766087720736709, 7.88502054404946264940649062912, 8.754617021155492037296087594307, 9.441406972246361776314869096725, 10.62020486219412429306225654015

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