Properties

Label 2-600-8.5-c1-0-8
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

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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

\(L(1)\) \(\approx\) \(0.170781 + 1.51964i\)
\(L(\frac12)\) \(\approx\) \(0.170781 + 1.51964i\)
\(L(\frac{3}{2})\) 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 \( 1 + 7.27T + 97T^{2} \)
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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

−11.04966961338269131631307783024, −10.10957022301563639240170964702, −9.421114710892092884881375134428, −8.376240233902104599818220807796, −7.20327985062965083877440358310, −6.51274375105669008042307614112, −5.59420562641212824519884618071, −4.55716357765669925569954760438, −3.52475047017929445879681177405, −2.70544020546026022641777625600, 0.62280680327511820602013929160, 2.53586987360629543266000101036, 3.26713608800385983286257918733, 4.47252596822288312361998375702, 5.82146562749990109371727854027, 6.55737072265480307788124034358, 7.05852679154603381357474675503, 8.891721650704399877276085034859, 9.328083766898338176692344501953, 10.51140794190026057118534973507

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