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Properties

Label 2-650-13.10-c1-0-20
Degree $2$
Conductor $650$
Sign $-0.822 - 0.568i$
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 + (−1.20 − 2.08i)3-s + (0.499 − 0.866i)4-s + (−2.08 − 1.20i)6-s + (−1.21 − 0.702i)7-s − 0.999i·8-s + (−1.40 + 2.43i)9-s + (−4.59 + 2.65i)11-s − 2.40·12-s + (−3.23 + 1.58i)13-s − 1.40·14-s + (−0.5 − 0.866i)16-s + (3.52 − 6.09i)17-s + 2.80i·18-s + (−2.28 − 1.32i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.695 − 1.20i)3-s + (0.249 − 0.433i)4-s + (−0.852 − 0.491i)6-s + (−0.460 − 0.265i)7-s − 0.353i·8-s + (−0.467 + 0.810i)9-s + (−1.38 + 0.800i)11-s − 0.695·12-s + (−0.897 + 0.440i)13-s − 0.375·14-s + (−0.125 − 0.216i)16-s + (0.853 − 1.47i)17-s + 0.661i·18-s + (−0.524 − 0.303i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.822 - 0.568i$
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: \((2,\ 650,\ (\ :1/2),\ -0.822 - 0.568i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.212671 + 0.682370i\)
\(L(\frac12)\) \(\approx\) \(0.212671 + 0.682370i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.23 - 1.58i)T \)
good3 \( 1 + (1.20 + 2.08i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.21 + 0.702i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.59 - 2.65i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.52 + 6.09i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.28 + 1.32i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.33 + 2.30i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.10 - 3.64i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.07iT - 31T^{2} \)
37 \( 1 + (0.715 - 0.412i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.43 + 1.40i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.64 + 4.57i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.79iT - 47T^{2} \)
53 \( 1 + 7.93T + 53T^{2} \)
59 \( 1 + (5.59 + 3.23i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.36 + 12.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.46 + 2i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.34 + 4.81i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.04iT - 73T^{2} \)
79 \( 1 + 4.05T + 79T^{2} \)
83 \( 1 - 8.55iT - 83T^{2} \)
89 \( 1 + (-13.1 + 7.57i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.78 + 2.18i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.23967904789616052366116564133, −9.436088069437700502470877344839, −7.85937156511762850442695468383, −7.14143970041256581691108240050, −6.57629933995300269632965253644, −5.33928061066976913452045750504, −4.74070790455411903498958354774, −3.01243841867521970766964035151, −1.99430084266109308285417743854, −0.31814063999806356760289433805, 2.70088159437688952015143815890, 3.77728299086243176324243201862, 4.72987494093584847058067365573, 5.78767264896900782640799352670, 5.93797383512144103940804798347, 7.62825261203928218213763685260, 8.301729455333318164522831871303, 9.656663314278849511838501287683, 10.29778785923466981037710875314, 10.91093928155290921528679460309

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