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Properties

Label 2-1274-91.25-c1-0-45
Degree $2$
Conductor $1274$
Sign $-0.921 - 0.389i$
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 + (1.65 − 2.86i)3-s + (0.499 − 0.866i)4-s + (−0.326 + 0.188i)5-s + 3.30i·6-s + 0.999i·8-s + (−3.96 − 6.86i)9-s + (0.188 − 0.326i)10-s + (−4.92 − 2.84i)11-s + (−1.65 − 2.86i)12-s + (−1.81 + 3.11i)13-s + 1.24i·15-s + (−0.5 − 0.866i)16-s + (0.623 − 1.07i)17-s + (6.86 + 3.96i)18-s + (−1.40 + 0.811i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.954 − 1.65i)3-s + (0.249 − 0.433i)4-s + (−0.145 + 0.0842i)5-s + 1.34i·6-s + 0.353i·8-s + (−1.32 − 2.28i)9-s + (0.0595 − 0.103i)10-s + (−1.48 − 0.856i)11-s + (−0.477 − 0.826i)12-s + (−0.502 + 0.864i)13-s + 0.321i·15-s + (−0.125 − 0.216i)16-s + (0.151 − 0.261i)17-s + (1.61 + 0.934i)18-s + (−0.322 + 0.186i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 - 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $-0.921 - 0.389i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ -0.921 - 0.389i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5825940044\)
\(L(\frac12)\) \(\approx\) \(0.5825940044\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.81 - 3.11i)T \)
good3 \( 1 + (-1.65 + 2.86i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.326 - 0.188i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.92 + 2.84i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.623 + 1.07i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.40 - 0.811i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.46 - 4.26i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.61T + 29T^{2} \)
31 \( 1 + (4.92 + 2.84i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.00 + 3.46i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.07iT - 41T^{2} \)
43 \( 1 - 3.24T + 43T^{2} \)
47 \( 1 + (-2.00 + 1.15i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.94 + 5.74i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.65 + 4.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.73 - 4.46i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.2iT - 71T^{2} \)
73 \( 1 + (-0.274 - 0.158i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.08 + 1.88i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.74iT - 83T^{2} \)
89 \( 1 + (8.00 - 4.62i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.9iT - 97T^{2} \)
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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

−9.107023995235801350221738866546, −8.115435513685994913440006217911, −7.62876094388195497976287645689, −7.17071934484849199909861055195, −6.14713503932218013908995930355, −5.38919279196030259662950372143, −3.56090384462617925610958284814, −2.56265960508895074356080848105, −1.70685436187244906638707648852, −0.24075071520922846894993722855, 2.38048302167817730783151003484, 2.86951279609440112692091432143, 4.04983138584754359471532618652, 4.78856908760493363460717737684, 5.63425191328502422054022051436, 7.39502807107004061083725773008, 7.979872975844548666357456422410, 8.644119190755754230440360585245, 9.463237140238317781951202847057, 10.12614026130546646651440054255

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