Properties

Label 2-600-8.5-c1-0-30
Degree $2$
Conductor $600$
Sign $-0.992 + 0.125i$
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  + (−0.0591 − 1.41i)2-s + i·3-s + (−1.99 + 0.167i)4-s + (1.41 − 0.0591i)6-s − 1.33·7-s + (0.353 + 2.80i)8-s − 9-s − 2.94i·11-s + (−0.167 − 1.99i)12-s − 2.04i·13-s + (0.0788 + 1.88i)14-s + (3.94 − 0.665i)16-s − 3.61·17-s + (0.0591 + 1.41i)18-s − 5.35i·19-s + ⋯
L(s)  = 1  + (−0.0418 − 0.999i)2-s + 0.577i·3-s + (−0.996 + 0.0835i)4-s + (0.576 − 0.0241i)6-s − 0.504·7-s + (0.125 + 0.992i)8-s − 0.333·9-s − 0.887i·11-s + (−0.0482 − 0.575i)12-s − 0.566i·13-s + (0.0210 + 0.503i)14-s + (0.986 − 0.166i)16-s − 0.876·17-s + (0.0139 + 0.333i)18-s − 1.22i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.992 + 0.125i$
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: \((2,\ 600,\ (\ :1/2),\ -0.992 + 0.125i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0344790 - 0.549016i\)
\(L(\frac12)\) \(\approx\) \(0.0344790 - 0.549016i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0591 + 1.41i)T \)
3 \( 1 - iT \)
5 \( 1 \)
good7 \( 1 + 1.33T + 7T^{2} \)
11 \( 1 + 2.94iT - 11T^{2} \)
13 \( 1 + 2.04iT - 13T^{2} \)
17 \( 1 + 3.61T + 17T^{2} \)
19 \( 1 + 5.35iT - 19T^{2} \)
23 \( 1 + 8.59T + 23T^{2} \)
29 \( 1 + 5.26iT - 29T^{2} \)
31 \( 1 + 2.08T + 31T^{2} \)
37 \( 1 - 6.55iT - 37T^{2} \)
41 \( 1 - 7.02T + 41T^{2} \)
43 \( 1 + 8.50iT - 43T^{2} \)
47 \( 1 + 9.97T + 47T^{2} \)
53 \( 1 + 6.12iT - 53T^{2} \)
59 \( 1 + 4.75iT - 59T^{2} \)
61 \( 1 - 8.51iT - 61T^{2} \)
67 \( 1 - 10.6iT - 67T^{2} \)
71 \( 1 + 2.62T + 71T^{2} \)
73 \( 1 - 15.3T + 73T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 - 1.52iT - 83T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 - 13.4T + 97T^{2} \)
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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

−10.22965682670011392470931379475, −9.614799362542995035744283444743, −8.717890024922613517394568054342, −8.000896277054957833219209629844, −6.42548837105786042684694005484, −5.41302932772298192960490374392, −4.32635461534388414763965068026, −3.40620100389946468601347042296, −2.36429255178007999394237042538, −0.30117491900860591649816704431, 1.86624157414278988443447680090, 3.70117816696416890318393086448, 4.71385474323476541732573721728, 5.98909546428856439326551573949, 6.56021009594169738357370446305, 7.50644234695293638164573007250, 8.186443926358151579258020999605, 9.292968901131230607190080301592, 9.855235021353804671253363259006, 11.01611822891837146835584901940

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