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Properties

Label 2-600-600.413-c1-0-95
Degree $2$
Conductor $600$
Sign $0.647 + 0.762i$
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  + (1.39 − 0.221i)2-s + (−0.786 + 1.54i)3-s + (1.90 − 0.618i)4-s + (−1.93 − 1.12i)5-s + (−0.756 + 2.32i)6-s + (3.72 − 3.72i)7-s + (2.52 − 1.28i)8-s + (−1.76 − 2.42i)9-s + (−2.94 − 1.14i)10-s + (−3.48 − 2.53i)11-s + (−0.541 + 3.42i)12-s + (4.38 − 6.03i)14-s + (3.25 − 2.09i)15-s + (3.23 − 2.35i)16-s + (−2.99 − 3i)18-s + ⋯
L(s)  = 1  + (0.987 − 0.156i)2-s + (−0.453 + 0.891i)3-s + (0.951 − 0.309i)4-s + (−0.863 − 0.503i)5-s + (−0.309 + 0.951i)6-s + (1.40 − 1.40i)7-s + (0.891 − 0.453i)8-s + (−0.587 − 0.809i)9-s + (−0.932 − 0.362i)10-s + (−1.05 − 0.763i)11-s + (−0.156 + 0.987i)12-s + (1.17 − 1.61i)14-s + (0.840 − 0.541i)15-s + (0.809 − 0.587i)16-s + (−0.707 − 0.707i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 + 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.647 + 0.762i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.647 + 0.762i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97375 - 0.913321i\)
\(L(\frac12)\) \(\approx\) \(1.97375 - 0.913321i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.221i)T \)
3 \( 1 + (0.786 - 1.54i)T \)
5 \( 1 + (1.93 + 1.12i)T \)
good7 \( 1 + (-3.72 + 3.72i)T - 7iT^{2} \)
11 \( 1 + (3.48 + 2.53i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (21.8 - 7.10i)T^{2} \)
29 \( 1 + (-8.89 + 2.88i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.40 - 10.4i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (35.1 + 11.4i)T^{2} \)
41 \( 1 + (-12.6 + 38.9i)T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (-27.6 - 38.0i)T^{2} \)
53 \( 1 + (2.50 - 4.90i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (-5.61 - 7.72i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (39.3 - 54.2i)T^{2} \)
71 \( 1 + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.464 + 2.93i)T + (-69.4 + 22.5i)T^{2} \)
79 \( 1 + (5.71 - 1.85i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (13.1 - 6.68i)T + (48.7 - 67.1i)T^{2} \)
89 \( 1 + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-6.33 - 3.22i)T + (57.0 + 78.4i)T^{2} \)
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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

−10.63853370142658048873414692725, −10.34360044970216043700425220333, −8.592330132285158646472188292981, −7.84970033893349963681662023553, −6.89550306795443290327417643838, −5.45637378597326534146631625530, −4.78179625251166293800024412009, −4.18087059273535018442586525706, −3.17419638630023707731814996253, −0.995648418896374232789093690326, 2.00817711037169617720739150490, 2.77183182461067812462264464172, 4.55821375408455091607584557789, 5.22805457061458657094229977902, 6.12724537088375012160055753935, 7.23510287636725092999554578765, 7.900480190320614053381661806664, 8.461609023712259972448505168626, 10.39516706084236949888702895502, 11.36099336465699744959585035645

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