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Properties

Label 2-600-120.59-c1-0-37
Degree $2$
Conductor $600$
Sign $0.344 - 0.938i$
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  + (1.29 + 0.578i)2-s + (−1.56 + 0.751i)3-s + (1.33 + 1.49i)4-s + (−2.44 + 0.0670i)6-s + 4.28·7-s + (0.852 + 2.69i)8-s + (1.86 − 2.34i)9-s − 2.44i·11-s + (−3.19 − 1.33i)12-s + 2.71·13-s + (5.53 + 2.48i)14-s + (−0.460 + 3.97i)16-s + 1.16·17-s + (3.77 − 1.94i)18-s − 6.05·19-s + ⋯
L(s)  = 1  + (0.912 + 0.409i)2-s + (−0.900 + 0.433i)3-s + (0.665 + 0.746i)4-s + (−0.999 + 0.0273i)6-s + 1.61·7-s + (0.301 + 0.953i)8-s + (0.623 − 0.781i)9-s − 0.737i·11-s + (−0.923 − 0.384i)12-s + 0.752·13-s + (1.47 + 0.662i)14-s + (−0.115 + 0.993i)16-s + 0.282·17-s + (0.888 − 0.458i)18-s − 1.38·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.344 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.344 - 0.938i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.344 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87510 + 1.30879i\)
\(L(\frac12)\) \(\approx\) \(1.87510 + 1.30879i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 - 0.578i)T \)
3 \( 1 + (1.56 - 0.751i)T \)
5 \( 1 \)
good7 \( 1 - 4.28T + 7T^{2} \)
11 \( 1 + 2.44iT - 11T^{2} \)
13 \( 1 - 2.71T + 13T^{2} \)
17 \( 1 - 1.16T + 17T^{2} \)
19 \( 1 + 6.05T + 19T^{2} \)
23 \( 1 - 7.55iT - 23T^{2} \)
29 \( 1 - 0.733T + 29T^{2} \)
31 \( 1 - 0.469iT - 31T^{2} \)
37 \( 1 + 1.36T + 37T^{2} \)
41 \( 1 + 4.69iT - 41T^{2} \)
43 \( 1 + 1.50iT - 43T^{2} \)
47 \( 1 + 4.07iT - 47T^{2} \)
53 \( 1 + 1.00iT - 53T^{2} \)
59 \( 1 + 1.63iT - 59T^{2} \)
61 \( 1 + 10.9iT - 61T^{2} \)
67 \( 1 - 9.97iT - 67T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 + 9.63iT - 73T^{2} \)
79 \( 1 - 3.61iT - 79T^{2} \)
83 \( 1 + 5.45T + 83T^{2} \)
89 \( 1 - 7.75iT - 89T^{2} \)
97 \( 1 + 17.1iT - 97T^{2} \)
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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

−11.16349408507166084748624215388, −10.42055900512696961965880151744, −8.833145980271914278690174033811, −8.085422054221720322434205539246, −7.05892868922254457045985052942, −5.96230648954543220932916904356, −5.36066068937268269112828957759, −4.44768442352260568815648241716, −3.57556353600361567001637445914, −1.67589276211129804961176080442, 1.31685093124457451312465225366, 2.30268909714447250067437783516, 4.32808160571140347571031861504, 4.71599768078233184735021675249, 5.81822605596500699759876664637, 6.62598519879820891623675619434, 7.62817922304770955822383446043, 8.589753125431769069005479070124, 10.21009540696666062729476048330, 10.78989941486644500113816143351

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