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Properties

Label 2-600-24.11-c1-0-55
Degree $2$
Conductor $600$
Sign $-0.964 + 0.264i$
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  + (0.814 − 1.15i)2-s + (−1.48 + 0.887i)3-s + (−0.672 − 1.88i)4-s + (−0.185 + 2.44i)6-s + 0.797i·7-s + (−2.72 − 0.757i)8-s + (1.42 − 2.64i)9-s − 0.320i·11-s + (2.67 + 2.20i)12-s − 4.30i·13-s + (0.921 + 0.649i)14-s + (−3.09 + 2.53i)16-s − 2.57i·17-s + (−1.89 − 3.79i)18-s − 6.10·19-s + ⋯
L(s)  = 1  + (0.576 − 0.817i)2-s + (−0.858 + 0.512i)3-s + (−0.336 − 0.941i)4-s + (−0.0756 + 0.997i)6-s + 0.301i·7-s + (−0.963 − 0.267i)8-s + (0.474 − 0.880i)9-s − 0.0966i·11-s + (0.771 + 0.636i)12-s − 1.19i·13-s + (0.246 + 0.173i)14-s + (−0.773 + 0.633i)16-s − 0.624i·17-s + (−0.446 − 0.894i)18-s − 1.40·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.264i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.964 + 0.264i$
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: \((2,\ 600,\ (\ :1/2),\ -0.964 + 0.264i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.108837 - 0.809844i\)
\(L(\frac12)\) \(\approx\) \(0.108837 - 0.809844i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.814 + 1.15i)T \)
3 \( 1 + (1.48 - 0.887i)T \)
5 \( 1 \)
good7 \( 1 - 0.797iT - 7T^{2} \)
11 \( 1 + 0.320iT - 11T^{2} \)
13 \( 1 + 4.30iT - 13T^{2} \)
17 \( 1 + 2.57iT - 17T^{2} \)
19 \( 1 + 6.10T + 19T^{2} \)
23 \( 1 + 3.13T + 23T^{2} \)
29 \( 1 + 8.79T + 29T^{2} \)
31 \( 1 + 9.90iT - 31T^{2} \)
37 \( 1 + 8.49iT - 37T^{2} \)
41 \( 1 - 5.28iT - 41T^{2} \)
43 \( 1 - 2.97T + 43T^{2} \)
47 \( 1 - 6.56T + 47T^{2} \)
53 \( 1 + 3.94T + 53T^{2} \)
59 \( 1 - 12.4iT - 59T^{2} \)
61 \( 1 + 8.83iT - 61T^{2} \)
67 \( 1 + 4.66T + 67T^{2} \)
71 \( 1 - 3.43T + 71T^{2} \)
73 \( 1 - 1.43T + 73T^{2} \)
79 \( 1 - 2.89iT - 79T^{2} \)
83 \( 1 + 3.37iT - 83T^{2} \)
89 \( 1 + 13.7iT - 89T^{2} \)
97 \( 1 - 4.26T + 97T^{2} \)
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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

−10.53417513186331574489743277323, −9.681982585140908190259668444378, −8.919226849375183428892470150976, −7.50086402260572548447219889014, −6.01692764784716983257942804134, −5.69595588111849232763105435475, −4.52507849625084041032553400038, −3.68491493868117623848587460239, −2.29538966048182734726224853324, −0.40302806519962244708462208222, 1.97935272395138734251722380847, 3.87164888483171023890551224340, 4.68237050256618351629640613917, 5.76302970847305692000859776330, 6.57091258523375248995744532628, 7.17058690550276661929104097167, 8.176133946073930490469949910546, 9.078170594209124675113961003061, 10.38672558728270444126013447019, 11.22260482396671818237223074423

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