Properties

Label 2-600-40.29-c1-0-6
Degree $2$
Conductor $600$
Sign $0.772 - 0.634i$
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  + (−0.796 − 1.16i)2-s + 3-s + (−0.731 + 1.86i)4-s + (−0.796 − 1.16i)6-s + 4.72i·7-s + (2.75 − 0.627i)8-s + 9-s − 3.93i·11-s + (−0.731 + 1.86i)12-s − 3.46·13-s + (5.51 − 3.76i)14-s + (−2.93 − 2.72i)16-s + 3.51i·17-s + (−0.796 − 1.16i)18-s + 5.44i·19-s + ⋯
L(s)  = 1  + (−0.563 − 0.826i)2-s + 0.577·3-s + (−0.365 + 0.930i)4-s + (−0.325 − 0.477i)6-s + 1.78i·7-s + (0.975 − 0.221i)8-s + 0.333·9-s − 1.18i·11-s + (−0.211 + 0.537i)12-s − 0.961·13-s + (1.47 − 1.00i)14-s + (−0.732 − 0.680i)16-s + 0.852i·17-s + (−0.187 − 0.275i)18-s + 1.24i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 - 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.772 - 0.634i$
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: \((2,\ 600,\ (\ :1/2),\ 0.772 - 0.634i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05553 + 0.377825i\)
\(L(\frac12)\) \(\approx\) \(1.05553 + 0.377825i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.796 + 1.16i)T \)
3 \( 1 - T \)
5 \( 1 \)
good7 \( 1 - 4.72iT - 7T^{2} \)
11 \( 1 + 3.93iT - 11T^{2} \)
13 \( 1 + 3.46T + 13T^{2} \)
17 \( 1 - 3.51iT - 17T^{2} \)
19 \( 1 - 5.44iT - 19T^{2} \)
23 \( 1 - 7.11iT - 23T^{2} \)
29 \( 1 - 3.66iT - 29T^{2} \)
31 \( 1 - 5.23T + 31T^{2} \)
37 \( 1 - 0.414T + 37T^{2} \)
41 \( 1 - 3.00T + 41T^{2} \)
43 \( 1 - 5.34T + 43T^{2} \)
47 \( 1 + 0.925iT - 47T^{2} \)
53 \( 1 + 0.233T + 53T^{2} \)
59 \( 1 + 14.3iT - 59T^{2} \)
61 \( 1 + 0.118iT - 61T^{2} \)
67 \( 1 + 13.4T + 67T^{2} \)
71 \( 1 - 2.19T + 71T^{2} \)
73 \( 1 - 0.563iT - 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 + 8.88T + 89T^{2} \)
97 \( 1 - 7.27iT - 97T^{2} \)
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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

−10.71401265428905046469263906538, −9.698272498844168564381436785469, −9.098821884025004954476814680971, −8.321833294709126184270493397968, −7.74323820052634336905524694511, −6.17217326917818870681427982343, −5.17929604465757827894595118964, −3.65401110249242816254299342341, −2.79445272217427513918392674720, −1.74808743734240588696126617749, 0.71989644200762941532033453472, 2.47916611851055270062720256990, 4.44599329893579208794227444084, 4.65802656610525912037347890212, 6.49954007998201195852845406426, 7.35152329066104432635264410570, 7.51970081355223431845068379030, 8.796978786736197962753822856328, 9.745993031381776759887104758159, 10.17513441834611318770602620148

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