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Properties

Label 2-600-24.11-c1-0-27
Degree $2$
Conductor $600$
Sign $0.142 + 0.989i$
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  + (−0.578 − 1.29i)2-s + (−0.751 − 1.56i)3-s + (−1.33 + 1.49i)4-s + (−1.57 + 1.87i)6-s + 4.28i·7-s + (2.69 + 0.852i)8-s + (−1.86 + 2.34i)9-s − 2.44i·11-s + (3.33 + 0.953i)12-s − 2.71i·13-s + (5.53 − 2.48i)14-s + (−0.460 − 3.97i)16-s − 1.16i·17-s + (4.10 + 1.05i)18-s + 6.05·19-s + ⋯
L(s)  = 1  + (−0.409 − 0.912i)2-s + (−0.433 − 0.900i)3-s + (−0.665 + 0.746i)4-s + (−0.644 + 0.764i)6-s + 1.61i·7-s + (0.953 + 0.301i)8-s + (−0.623 + 0.781i)9-s − 0.737i·11-s + (0.961 + 0.275i)12-s − 0.752i·13-s + (1.47 − 0.662i)14-s + (−0.115 − 0.993i)16-s − 0.282i·17-s + (0.968 + 0.248i)18-s + 1.38·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.142 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.142 + 0.989i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.142 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.721457 - 0.625187i\)
\(L(\frac12)\) \(\approx\) \(0.721457 - 0.625187i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−10.77139245261315440709474001341, −9.495211815752524146086744354752, −8.768066828231365827760839487635, −8.043891014609381342263136540746, −7.05486221646182160661298743910, −5.65283627325455011365436566029, −5.17325085592740241063480971461, −3.17530034890267283243443633924, −2.44157132438489794531157485235, −0.935040670615472837041616410840, 1.00353715825309894413699242036, 3.62451022523236626755006110770, 4.55039927776453661307668987769, 5.22721069513107719900397919528, 6.61609847447747067998020906971, 7.12398550793156299080692059682, 8.142629544453771930397533512177, 9.444034127363808249714774968068, 9.717821331019544404614716760054, 10.72100679368882157878377452509

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