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Properties

Label 2-600-8.5-c1-0-10
Degree $2$
Conductor $600$
Sign $0.302 - 0.953i$
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  + (−1.40 − 0.144i)2-s + i·3-s + (1.95 + 0.406i)4-s + (0.144 − 1.40i)6-s + 3.62·7-s + (−2.69 − 0.855i)8-s − 9-s + 6.20i·11-s + (−0.406 + 1.95i)12-s + 0.578i·13-s + (−5.10 − 0.524i)14-s + (3.66 + 1.59i)16-s − 1.42·17-s + (1.40 + 0.144i)18-s − 5.62i·19-s + ⋯
L(s)  = 1  + (−0.994 − 0.102i)2-s + 0.577i·3-s + (0.979 + 0.203i)4-s + (0.0590 − 0.574i)6-s + 1.37·7-s + (−0.953 − 0.302i)8-s − 0.333·9-s + 1.87i·11-s + (−0.117 + 0.565i)12-s + 0.160i·13-s + (−1.36 − 0.140i)14-s + (0.917 + 0.398i)16-s − 0.344·17-s + (0.331 + 0.0340i)18-s − 1.29i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.302 - 0.953i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.302 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.844521 + 0.618052i\)
\(L(\frac12)\) \(\approx\) \(0.844521 + 0.618052i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−10.89676938187121560673782679705, −9.827301003361831681847558985509, −9.216001210864604849294216656828, −8.340686412981253728349330453208, −7.42025464417714180457156283080, −6.71501099131561653085446243717, −5.09491099169146447756957214220, −4.44402680576877082698655176151, −2.70964227011762655971627405324, −1.55453057656037348169259121235, 0.873423304839546583199217713719, 2.08732678671105850208355199259, 3.48783034461214496032036352762, 5.32915562520870735439483302970, 6.04391569230839519112243752601, 7.17065160887224147619728458966, 8.140394407826987335262403315633, 8.417776423194298973240818501908, 9.402815800720190397725683938410, 10.89861549090212939331677662967

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