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Properties

Label 2-650-65.8-c1-0-12
Degree $2$
Conductor $650$
Sign $0.661 - 0.749i$
Analytic cond. $5.19027$
Root an. cond. $2.27821$
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  + 2-s + (2 + 2i)3-s + 4-s + (2 + 2i)6-s − 4i·7-s + 8-s + 5i·9-s + (−2 + 2i)11-s + (2 + 2i)12-s + (2 + 3i)13-s − 4i·14-s + 16-s + (3 + 3i)17-s + 5i·18-s + (−2 + 2i)19-s + ⋯
L(s)  = 1  + 0.707·2-s + (1.15 + 1.15i)3-s + 0.5·4-s + (0.816 + 0.816i)6-s − 1.51i·7-s + 0.353·8-s + 1.66i·9-s + (−0.603 + 0.603i)11-s + (0.577 + 0.577i)12-s + (0.554 + 0.832i)13-s − 1.06i·14-s + 0.250·16-s + (0.727 + 0.727i)17-s + 1.17i·18-s + (−0.458 + 0.458i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.661 - 0.749i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.661 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.94270 + 1.32752i\)
\(L(\frac12)\) \(\approx\) \(2.94270 + 1.32752i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 \)
13 \( 1 + (-2 - 3i)T \)
good3 \( 1 + (-2 - 2i)T + 3iT^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + (2 - 2i)T - 11iT^{2} \)
17 \( 1 + (-3 - 3i)T + 17iT^{2} \)
19 \( 1 + (2 - 2i)T - 19iT^{2} \)
23 \( 1 + (-2 + 2i)T - 23iT^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 + (6 + 6i)T + 31iT^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + (1 + i)T + 41iT^{2} \)
43 \( 1 + (2 - 2i)T - 43iT^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 + (-1 - i)T + 53iT^{2} \)
59 \( 1 + (2 + 2i)T + 59iT^{2} \)
61 \( 1 + 12T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (2 + 2i)T + 71iT^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 + 8iT - 83T^{2} \)
89 \( 1 + (-5 - 5i)T + 89iT^{2} \)
97 \( 1 + 97T^{2} \)
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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

−10.45482610981743914304423159499, −10.03842060204705412044743419712, −9.034890958930216718659071302942, −7.955494208720469956249099425281, −7.34215368291357507670157760830, −6.05102248518211326996855072120, −4.63249321518354774566911557141, −4.08156166234688855214334650344, −3.40244502515591778640501783780, −1.98890084549157337313457793293, 1.57673875979509692122494613027, 2.92726761012816273224139569727, 3.12778053219394646716512195739, 5.15637704471511326845205558602, 5.86941378321270130757415554123, 6.93901187287315527112138887520, 7.81668272242571049085112115960, 8.607476713993641986366879919145, 9.149582604927927579105012583059, 10.58678052830156172268302830916

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