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Properties

Label 2-600-600.53-c1-0-102
Degree $2$
Conductor $600$
Sign $-0.837 - 0.546i$
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  + (−0.642 − 1.26i)2-s + (−0.270 − 1.71i)3-s + (−1.17 + 1.61i)4-s + (−2.04 − 0.901i)5-s + (−1.98 + 1.43i)6-s + (3.64 − 3.64i)7-s + (2.79 + 0.442i)8-s + (−2.85 + 0.927i)9-s + (0.178 + 3.15i)10-s + (2.04 − 6.28i)11-s + (3.08 + 1.57i)12-s + (−6.92 − 2.25i)14-s + (−0.987 + 3.74i)15-s + (−1.23 − 3.80i)16-s + (3 + 3i)18-s + ⋯
L(s)  = 1  + (−0.453 − 0.891i)2-s + (−0.156 − 0.987i)3-s + (−0.587 + 0.809i)4-s + (−0.915 − 0.403i)5-s + (−0.809 + 0.587i)6-s + (1.37 − 1.37i)7-s + (0.987 + 0.156i)8-s + (−0.951 + 0.309i)9-s + (0.0563 + 0.998i)10-s + (0.616 − 1.89i)11-s + (0.891 + 0.453i)12-s + (−1.85 − 0.601i)14-s + (−0.254 + 0.966i)15-s + (−0.309 − 0.951i)16-s + (0.707 + 0.707i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.837 - 0.546i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.837 - 0.546i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.253539 + 0.852195i\)
\(L(\frac12)\) \(\approx\) \(0.253539 + 0.852195i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.642 + 1.26i)T \)
3 \( 1 + (0.270 + 1.71i)T \)
5 \( 1 + (2.04 + 0.901i)T \)
good7 \( 1 + (-3.64 + 3.64i)T - 7iT^{2} \)
11 \( 1 + (-2.04 + 6.28i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-13.5 + 18.6i)T^{2} \)
29 \( 1 + (3.14 - 4.32i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.54 - 1.12i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-21.7 - 29.9i)T^{2} \)
41 \( 1 + (33.1 - 24.0i)T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (-1.93 - 12.2i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (-3.92 + 1.27i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (63.7 + 20.7i)T^{2} \)
71 \( 1 + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-14.9 + 7.63i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (2.92 - 4.01i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (14.5 + 2.30i)T + (78.9 + 25.6i)T^{2} \)
89 \( 1 + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-1.07 + 0.170i)T + (92.2 - 29.9i)T^{2} \)
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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

−10.69976041665569707810378308174, −8.968314332576499672146716675881, −8.332706862578844815756709467914, −7.75679788307864829796697811355, −6.97017535032242859194588083162, −5.35168807127513379765900401503, −4.18492997305120955816560563855, −3.29506430187147601208744440527, −1.44612413204244568352432077915, −0.66339063236561570594431833908, 2.11609993602496549068607484901, 4.09579381221170769835006824378, 4.77458796319452723223282211857, 5.58649554460552812743678882245, 6.80894380402975494466592553976, 7.80962694723095942494388399940, 8.511301091053120741879055851069, 9.338570535844305831251906546200, 10.07304550153865508961366701942, 11.19659696377271045391119708735

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