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Properties

Label 2-1274-91.88-c1-0-5
Degree $2$
Conductor $1274$
Sign $-0.997 - 0.0719i$
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 + 0.252·3-s + (0.499 + 0.866i)4-s + (−0.993 + 0.573i)5-s + (0.218 + 0.126i)6-s + 0.999i·8-s − 2.93·9-s − 1.14·10-s + 4.44i·11-s + (0.126 + 0.218i)12-s + (−3.54 − 0.666i)13-s + (−0.251 + 0.145i)15-s + (−0.5 + 0.866i)16-s + (1.35 + 2.34i)17-s + (−2.54 − 1.46i)18-s − 6.56i·19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + 0.145·3-s + (0.249 + 0.433i)4-s + (−0.444 + 0.256i)5-s + (0.0894 + 0.0516i)6-s + 0.353i·8-s − 0.978·9-s − 0.362·10-s + 1.34i·11-s + (0.0364 + 0.0632i)12-s + (−0.982 − 0.184i)13-s + (−0.0649 + 0.0374i)15-s + (−0.125 + 0.216i)16-s + (0.328 + 0.569i)17-s + (−0.599 − 0.346i)18-s − 1.50i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $-0.997 - 0.0719i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ -0.997 - 0.0719i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9203321678\)
\(L(\frac12)\) \(\approx\) \(0.9203321678\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.54 + 0.666i)T \)
good3 \( 1 - 0.252T + 3T^{2} \)
5 \( 1 + (0.993 - 0.573i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 4.44iT - 11T^{2} \)
17 \( 1 + (-1.35 - 2.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 6.56iT - 19T^{2} \)
23 \( 1 + (1.03 - 1.80i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.59 + 6.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.84 + 3.95i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.35 - 4.82i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (8.22 - 4.74i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.70 - 2.94i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.45 - 0.837i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.64 - 11.5i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.0586 - 0.0338i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 8.10T + 61T^{2} \)
67 \( 1 + 0.513iT - 67T^{2} \)
71 \( 1 + (-9.34 - 5.39i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.94 - 4.01i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.37 + 9.30i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.3iT - 83T^{2} \)
89 \( 1 + (9.40 + 5.42i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.84 - 1.06i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.859967037746160563200877183803, −9.371913165057434691893276569940, −8.128681577945858171884353264233, −7.57928013985405842770315993587, −6.86168784803614200107167228832, −5.83363114579229411996863804935, −4.97277420340364428461847458352, −4.14710502247024609580809250618, −3.05079003986273915712482108542, −2.15361973320404360736414705988, 0.28032358362609541551281229577, 2.02580035510653242173950873699, 3.23348228264193901142961389718, 3.80483974821901665395361187206, 5.15814483073716873559246939309, 5.63845714275383757213078603584, 6.66503744945415216082284102716, 7.77135785030815684979421301076, 8.433874661954883600259529784181, 9.274471799495893606765069813936

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