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Properties

Label 2-650-13.10-c1-0-8
Degree $2$
Conductor $650$
Sign $0.528 - 0.848i$
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  + (0.866 − 0.5i)2-s + (1.53 + 2.65i)3-s + (0.499 − 0.866i)4-s + (2.65 + 1.53i)6-s + (3.20 + 1.84i)7-s − 0.999i·8-s + (−3.21 + 5.56i)9-s + (−0.924 + 0.533i)11-s + 3.07·12-s + (−1.73 − 3.15i)13-s + 3.69·14-s + (−0.5 − 0.866i)16-s + (1.39 − 2.42i)17-s + 6.43i·18-s + (−1.90 − 1.09i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.886 + 1.53i)3-s + (0.249 − 0.433i)4-s + (1.08 + 0.626i)6-s + (1.20 + 0.698i)7-s − 0.353i·8-s + (−1.07 + 1.85i)9-s + (−0.278 + 0.160i)11-s + 0.886·12-s + (−0.481 − 0.876i)13-s + 0.987·14-s + (−0.125 − 0.216i)16-s + (0.339 − 0.587i)17-s + 1.51i·18-s + (−0.436 − 0.252i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.528 - 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.528 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.528 - 0.848i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.528 - 0.848i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.56951 + 1.42615i\)
\(L(\frac12)\) \(\approx\) \(2.56951 + 1.42615i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (1.73 + 3.15i)T \)
good3 \( 1 + (-1.53 - 2.65i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-3.20 - 1.84i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.924 - 0.533i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.39 + 2.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.90 + 1.09i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.565 - 0.979i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.36 - 2.36i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.20iT - 31T^{2} \)
37 \( 1 + (8.93 - 5.15i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.27 + 2.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.19 + 3.80i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.5iT - 47T^{2} \)
53 \( 1 - 3.81T + 53T^{2} \)
59 \( 1 + (-3.12 - 1.80i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.61 - 13.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.46 + 2i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-10.6 - 6.14i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.23iT - 73T^{2} \)
79 \( 1 + 14.2T + 79T^{2} \)
83 \( 1 + 8.18iT - 83T^{2} \)
89 \( 1 + (-7.62 + 4.40i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.57 - 4.37i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.52608308867658663842925958529, −10.04953918459771147268470144278, −8.983807820395194626424275257138, −8.381492042102163359082087013967, −7.38126956824175059405966692271, −5.47779957357953914620455727035, −5.08668259035927512606621116447, −4.18157899728010312736687770265, −3.06149284528766827973142689974, −2.20923416652363327287978150869, 1.42952405576771819065329621109, 2.40111620137900872934288802922, 3.74511969650728100709710271526, 4.86268205970724589436848555662, 6.18444166415871751773281530229, 6.99318397473634166301409320824, 7.76059621822638784363576131950, 8.208723515719892739985734722312, 9.157180235862271964373891972803, 10.64714399086995947159213554557

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