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Properties

Label 2-650-25.16-c1-0-5
Degree $2$
Conductor $650$
Sign $-0.956 + 0.292i$
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  + (0.809 + 0.587i)2-s + (0.925 + 2.84i)3-s + (0.309 + 0.951i)4-s + (−1.66 − 1.49i)5-s + (−0.925 + 2.84i)6-s − 3.79·7-s + (−0.309 + 0.951i)8-s + (−4.83 + 3.51i)9-s + (−0.469 − 2.18i)10-s + (0.674 + 0.490i)11-s + (−2.42 + 1.76i)12-s + (0.809 − 0.587i)13-s + (−3.07 − 2.23i)14-s + (2.71 − 6.12i)15-s + (−0.809 + 0.587i)16-s + (−1.23 + 3.81i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.534 + 1.64i)3-s + (0.154 + 0.475i)4-s + (−0.744 − 0.667i)5-s + (−0.377 + 1.16i)6-s − 1.43·7-s + (−0.109 + 0.336i)8-s + (−1.61 + 1.17i)9-s + (−0.148 − 0.691i)10-s + (0.203 + 0.147i)11-s + (−0.699 + 0.508i)12-s + (0.224 − 0.163i)13-s + (−0.820 − 0.596i)14-s + (0.700 − 1.58i)15-s + (−0.202 + 0.146i)16-s + (−0.300 + 0.925i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.956 + 0.292i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ -0.956 + 0.292i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.188601 - 1.26024i\)
\(L(\frac12)\) \(\approx\) \(0.188601 - 1.26024i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (1.66 + 1.49i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (-0.925 - 2.84i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + 3.79T + 7T^{2} \)
11 \( 1 + (-0.674 - 0.490i)T + (3.39 + 10.4i)T^{2} \)
17 \( 1 + (1.23 - 3.81i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.471 - 1.44i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (4.15 + 3.01i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.30 - 7.09i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.254 - 0.782i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.288 - 0.209i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.84 + 3.52i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 6.99T + 43T^{2} \)
47 \( 1 + (0.471 + 1.45i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.50 - 10.7i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.420 + 0.305i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-11.4 - 8.33i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.74 - 8.43i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (3.24 + 9.98i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (12.1 + 8.83i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.39 + 7.36i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.02 - 6.24i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-11.1 - 8.10i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-5.12 - 15.7i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.73073240502082675859182920542, −10.20066154079190193836080853446, −9.046170493275370221978982878520, −8.752584615674140906454848386697, −7.63401469801438262658826046919, −6.34188892191475988573333557641, −5.39994606916419375694723823513, −4.21723671496265788476972890598, −3.86542618384766533533676293962, −2.88884822584651429296405490779, 0.51052454950118445178715951548, 2.34006292144564508794835021354, 3.06750634929521965497408927903, 4.02996355370765609533418446898, 5.94321735110715372249069042882, 6.59337699386899201191169202593, 7.21293590729376941550983145851, 8.092974114787699915602154883230, 9.207141763642038394234666848385, 10.09517725561544332889673481138

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