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Properties

Label 2-600-40.29-c1-0-33
Degree $2$
Conductor $600$
Sign $-0.714 + 0.700i$
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  + (0.576 − 1.29i)2-s + 3-s + (−1.33 − 1.48i)4-s + (0.576 − 1.29i)6-s − 1.97i·7-s + (−2.69 + 0.867i)8-s + 9-s − 1.43i·11-s + (−1.33 − 1.48i)12-s + 0.241·13-s + (−2.55 − 1.13i)14-s + (−0.430 + 3.97i)16-s − 7.38i·17-s + (0.576 − 1.29i)18-s − 3.04i·19-s + ⋯
L(s)  = 1  + (0.407 − 0.913i)2-s + 0.577·3-s + (−0.667 − 0.744i)4-s + (0.235 − 0.527i)6-s − 0.747i·7-s + (−0.951 + 0.306i)8-s + 0.333·9-s − 0.431i·11-s + (−0.385 − 0.429i)12-s + 0.0669·13-s + (−0.682 − 0.304i)14-s + (−0.107 + 0.994i)16-s − 1.79i·17-s + (0.135 − 0.304i)18-s − 0.697i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.714 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.714 + 0.700i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.714 + 0.700i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.707213 - 1.73164i\)
\(L(\frac12)\) \(\approx\) \(0.707213 - 1.73164i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.576 + 1.29i)T \)
3 \( 1 - T \)
5 \( 1 \)
good7 \( 1 + 1.97iT - 7T^{2} \)
11 \( 1 + 1.43iT - 11T^{2} \)
13 \( 1 - 0.241T + 13T^{2} \)
17 \( 1 + 7.38iT - 17T^{2} \)
19 \( 1 + 3.04iT - 19T^{2} \)
23 \( 1 + 0.874iT - 23T^{2} \)
29 \( 1 - 9.07iT - 29T^{2} \)
31 \( 1 + 7.44T + 31T^{2} \)
37 \( 1 - 8.81T + 37T^{2} \)
41 \( 1 + 1.91T + 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 + 3.34iT - 47T^{2} \)
53 \( 1 + 9.20T + 53T^{2} \)
59 \( 1 - 6.43iT - 59T^{2} \)
61 \( 1 - 4.57iT - 61T^{2} \)
67 \( 1 - 4.86T + 67T^{2} \)
71 \( 1 + 8.21T + 71T^{2} \)
73 \( 1 - 4.12iT - 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 - 12.3T + 83T^{2} \)
89 \( 1 - 8.08T + 89T^{2} \)
97 \( 1 - 10.6iT - 97T^{2} \)
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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

−10.56641605948318541353578770001, −9.355043950314233074502443926339, −9.052926641764565143307775305225, −7.69766404928039738858623983929, −6.79979835662552747115252955464, −5.41312375999659660748266670470, −4.48677622717163763341784272267, −3.44025775314523136008444902405, −2.51770187819489101504788724970, −0.898075358576082711519951453598, 2.16118198385817224865390520683, 3.58095097182615840034577816759, 4.42267411532261301760436902602, 5.75123259890916661096552741192, 6.30765011514656755646882831932, 7.66599742671203865340234246213, 8.111843089841353291848359005474, 9.097660593851743474362919467335, 9.755759802302199973238472856898, 11.00303325565314721471618948823

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