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Properties

Label 2-1274-91.51-c1-0-19
Degree $2$
Conductor $1274$
Sign $-0.701 - 0.712i$
Analytic cond. $10.1729$
Root an. cond. $3.18950$
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 + (1.41 + 2.44i)3-s + (0.499 + 0.866i)4-s + (1.22 + 0.707i)5-s + 2.82i·6-s + 0.999i·8-s + (−2.49 + 4.33i)9-s + (0.707 + 1.22i)10-s + (−1.41 + 2.44i)12-s + (0.707 − 3.53i)13-s + 4i·15-s + (−0.5 + 0.866i)16-s + (0.707 + 1.22i)17-s + (−4.33 + 2.5i)18-s + (2.44 + 1.41i)19-s + 1.41i·20-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.816 + 1.41i)3-s + (0.249 + 0.433i)4-s + (0.547 + 0.316i)5-s + 1.15i·6-s + 0.353i·8-s + (−0.833 + 1.44i)9-s + (0.223 + 0.387i)10-s + (−0.408 + 0.707i)12-s + (0.196 − 0.980i)13-s + 1.03i·15-s + (−0.125 + 0.216i)16-s + (0.171 + 0.297i)17-s + (−1.02 + 0.589i)18-s + (0.561 + 0.324i)19-s + 0.316i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $-0.701 - 0.712i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ -0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.342412578\)
\(L(\frac12)\) \(\approx\) \(3.342412578\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 \)
13 \( 1 + (-0.707 + 3.53i)T \)
good3 \( 1 + (-1.41 - 2.44i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.22 - 0.707i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.707 - 1.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.44 - 1.41i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.66 - 5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 9.89iT - 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (9.79 + 5.65i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.44 - 1.41i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.36 + 11.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.3 + 6i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8iT - 71T^{2} \)
73 \( 1 + (-11.0 + 6.36i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.82iT - 83T^{2} \)
89 \( 1 + (-11.0 - 6.36i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.24iT - 97T^{2} \)
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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

−9.912978501544385579756420671376, −9.369578708343528571375286178459, −8.209452962043816143192347776668, −7.83190741002122038392869897618, −6.45225038156950963242284237709, −5.56591547042198176586935087114, −4.91465316984716100135226500110, −3.71792793352017503443647919394, −3.32399316787824613068718879915, −2.14440189917251789150360796320, 1.10119093408529921551549579478, 2.05682600198904807896158930183, 2.81400005175153150255452526776, 4.02393062780228445436694885106, 5.14920606569364800341929841084, 6.27930250262521506681244699502, 6.72702239271847694709511931675, 7.72673660991824414788334712357, 8.471579167513909317410027138224, 9.356443007221192495047495659100

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