↑

Properties

Label 2-600-8.5-c1-0-14
Degree $2$
Conductor $600$
Sign $0.975 + 0.221i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.16 − 0.796i)2-s − i·3-s + (0.731 + 1.86i)4-s + (−0.796 + 1.16i)6-s + 4.72·7-s + (0.627 − 2.75i)8-s − 9-s + 3.93i·11-s + (1.86 − 0.731i)12-s + 3.46i·13-s + (−5.51 − 3.76i)14-s + (−2.93 + 2.72i)16-s + 3.51·17-s + (1.16 + 0.796i)18-s + 5.44i·19-s + ⋯
L(s)  = 1  + (−0.826 − 0.563i)2-s − 0.577i·3-s + (0.365 + 0.930i)4-s + (−0.325 + 0.477i)6-s + 1.78·7-s + (0.221 − 0.975i)8-s − 0.333·9-s + 1.18i·11-s + (0.537 − 0.211i)12-s + 0.961i·13-s + (−1.47 − 1.00i)14-s + (−0.732 + 0.680i)16-s + 0.852·17-s + (0.275 + 0.187i)18-s + 1.24i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.975 + 0.221i$
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: \((2,\ 600,\ (\ :1/2),\ 0.975 + 0.221i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16782 - 0.131243i\)
\(L(\frac12)\) \(\approx\) \(1.16782 - 0.131243i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 + 0.796i)T \)
3 \( 1 + iT \)
5 \( 1 \)
good7 \( 1 - 4.72T + 7T^{2} \)
11 \( 1 - 3.93iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 - 3.51T + 17T^{2} \)
19 \( 1 - 5.44iT - 19T^{2} \)
23 \( 1 + 7.11T + 23T^{2} \)
29 \( 1 - 3.66iT - 29T^{2} \)
31 \( 1 - 5.23T + 31T^{2} \)
37 \( 1 - 0.414iT - 37T^{2} \)
41 \( 1 - 3.00T + 41T^{2} \)
43 \( 1 + 5.34iT - 43T^{2} \)
47 \( 1 + 0.925T + 47T^{2} \)
53 \( 1 - 0.233iT - 53T^{2} \)
59 \( 1 + 14.3iT - 59T^{2} \)
61 \( 1 - 0.118iT - 61T^{2} \)
67 \( 1 + 13.4iT - 67T^{2} \)
71 \( 1 - 2.19T + 71T^{2} \)
73 \( 1 + 0.563T + 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 + 11.3iT - 83T^{2} \)
89 \( 1 - 8.88T + 89T^{2} \)
97 \( 1 - 7.27T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.57649220167357949771149257686, −9.871063692452405005987920834955, −8.788497756789522624330892978556, −7.87128760985672337203627759422, −7.59386276537625650490350254291, −6.36989608041383603995884904983, −4.92037304635349676894395788740, −3.88625640498382999623210051292, −2.07495300574946916118790845721, −1.53770432509335201380842929261, 0.979232241270729640816128173531, 2.63731483401147211229509808676, 4.37232299991476007685130724664, 5.37398888021768623164291269505, 5.99999055058656260029215823377, 7.52462425355846172400414101610, 8.191981378554485948951283156414, 8.667331750639655198763320758778, 9.851851920729218998255099758654, 10.59914142760750098817167823139

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