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Properties

Label 2-1274-91.25-c1-0-23
Degree $2$
Conductor $1274$
Sign $0.653 - 0.756i$
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 + (−0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + (1.73 − i)5-s − 0.999i·6-s + 0.999i·8-s + (1 + 1.73i)9-s + (−0.999 + 1.73i)10-s + (4.33 + 2.5i)11-s + (0.499 + 0.866i)12-s + (3 − 2i)13-s + 1.99i·15-s + (−0.5 − 0.866i)16-s + (−1 + 1.73i)17-s + (−1.73 − i)18-s + (3.46 − 2i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s + (0.774 − 0.447i)5-s − 0.408i·6-s + 0.353i·8-s + (0.333 + 0.577i)9-s + (−0.316 + 0.547i)10-s + (1.30 + 0.753i)11-s + (0.144 + 0.249i)12-s + (0.832 − 0.554i)13-s + 0.516i·15-s + (−0.125 − 0.216i)16-s + (−0.242 + 0.420i)17-s + (−0.408 − 0.235i)18-s + (0.794 − 0.458i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 - 0.756i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.653 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $0.653 - 0.756i$
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: \((2,\ 1274,\ (\ :1/2),\ 0.653 - 0.756i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.492103822\)
\(L(\frac12)\) \(\approx\) \(1.492103822\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3 + 2i)T \)
good3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.73 + i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.33 - 2.5i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.46 + 2i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.5 + 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-4.33 - 2.5i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.59 - 1.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 5iT - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-11.2 + 6.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7 - 12.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.19 - 3i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.5 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.59 - 1.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (0.866 + 0.5i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.5 - 12.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + (5.19 - 3i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 7iT - 97T^{2} \)
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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

−9.737411801685598889211044968132, −9.059153208777203026190368539610, −8.356388161265102471638291628482, −7.32443099051038840191868457657, −6.43069064528753630204140200699, −5.71229523537798997911376689156, −4.78777986126239038279735444238, −3.93428050591016947574146427348, −2.21864160866540545587045206303, −1.16348064775005788398869821073, 1.04451311946762261412267215593, 1.86319657180959436432095696308, 3.31032114880092759666674612648, 4.08352473379743349453666624895, 5.87954195348273319911378416871, 6.22542977255707360494809971865, 7.05552699103440014361041191323, 7.934391569490228664054886574515, 9.064049454752289689220834236374, 9.483906162459854711072815867505

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