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Properties

Label 2-1274-91.4-c1-0-23
Degree $2$
Conductor $1274$
Sign $0.986 - 0.163i$
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  − i·2-s + (1.14 + 1.98i)3-s − 4-s + (−0.781 + 0.450i)5-s + (1.98 − 1.14i)6-s + i·8-s + (−1.12 + 1.94i)9-s + (0.450 + 0.781i)10-s + (3.75 − 2.16i)11-s + (−1.14 − 1.98i)12-s + (0.426 − 3.58i)13-s + (−1.78 − 1.03i)15-s + 16-s + 5.06·17-s + (1.94 + 1.12i)18-s + (5.34 + 3.08i)19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.661 + 1.14i)3-s − 0.5·4-s + (−0.349 + 0.201i)5-s + (0.809 − 0.467i)6-s + 0.353i·8-s + (−0.374 + 0.648i)9-s + (0.142 + 0.246i)10-s + (1.13 − 0.653i)11-s + (−0.330 − 0.572i)12-s + (0.118 − 0.992i)13-s + (−0.461 − 0.266i)15-s + 0.250·16-s + 1.22·17-s + (0.458 + 0.264i)18-s + (1.22 + 0.708i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $0.986 - 0.163i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ 0.986 - 0.163i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.055078288\)
\(L(\frac12)\) \(\approx\) \(2.055078288\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
7 \( 1 \)
13 \( 1 + (-0.426 + 3.58i)T \)
good3 \( 1 + (-1.14 - 1.98i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.781 - 0.450i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.75 + 2.16i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 5.06T + 17T^{2} \)
19 \( 1 + (-5.34 - 3.08i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 8.45T + 23T^{2} \)
29 \( 1 + (-1.09 + 1.89i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.756 - 0.436i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.144iT - 37T^{2} \)
41 \( 1 + (-3.46 - 1.99i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.85 - 6.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.52 + 1.46i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.848 - 1.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.54iT - 59T^{2} \)
61 \( 1 + (-4.16 + 7.21i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.99 + 5.19i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.83 - 1.63i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.466 - 0.269i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.26 + 5.65i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.2iT - 83T^{2} \)
89 \( 1 + 7.79iT - 89T^{2} \)
97 \( 1 + (10.1 - 5.85i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.803006785923457993402265937263, −9.189094415297086291871781561385, −8.168513424478881188651213761919, −7.70952546367137090612923978602, −6.10210966652333742808270994841, −5.32555598136144007714212093134, −4.01396063880473473974301649033, −3.63185693475081065344101692765, −2.87959879045379512864224778670, −1.16909772479348582896497614302, 1.07064820318762530779162187721, 2.17754609589696503438356294295, 3.63379988955079008234820115192, 4.46116976350192695055383691670, 5.71713900605691380667908638220, 6.63920061015697217468280568743, 7.26225167165322523780672012889, 7.85190867240160894936609086372, 8.623449297085115617002467864942, 9.395029558870240438802519066690

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