Properties

Label 2-1274-91.88-c1-0-12
Degree $2$
Conductor $1274$
Sign $0.684 - 0.729i$
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 − 0.466·3-s + (0.499 + 0.866i)4-s + (2.93 − 1.69i)5-s + (0.404 + 0.233i)6-s − 0.999i·8-s − 2.78·9-s − 3.38·10-s + 0.822i·11-s + (−0.233 − 0.404i)12-s + (−2.74 + 2.33i)13-s + (−1.36 + 0.790i)15-s + (−0.5 + 0.866i)16-s + (2.29 + 3.96i)17-s + (2.40 + 1.39i)18-s + 5.90i·19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s − 0.269·3-s + (0.249 + 0.433i)4-s + (1.31 − 0.757i)5-s + (0.164 + 0.0952i)6-s − 0.353i·8-s − 0.927·9-s − 1.07·10-s + 0.248i·11-s + (−0.0673 − 0.116i)12-s + (−0.762 + 0.646i)13-s + (−0.353 + 0.204i)15-s + (−0.125 + 0.216i)16-s + (0.555 + 0.962i)17-s + (0.567 + 0.327i)18-s + 1.35i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.684 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $0.684 - 0.729i$
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: \((2,\ 1274,\ (\ :1/2),\ 0.684 - 0.729i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.037349853\)
\(L(\frac12)\) \(\approx\) \(1.037349853\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.74 - 2.33i)T \)
good3 \( 1 + 0.466T + 3T^{2} \)
5 \( 1 + (-2.93 + 1.69i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 0.822iT - 11T^{2} \)
17 \( 1 + (-2.29 - 3.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 5.90iT - 19T^{2} \)
23 \( 1 + (3.06 - 5.30i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.43 - 5.94i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.71 + 2.14i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.39 - 4.84i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0774 - 0.0446i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.67 + 6.36i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.67 - 5.58i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.50 + 6.07i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.50 - 0.870i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 2.37T + 61T^{2} \)
67 \( 1 - 0.291iT - 67T^{2} \)
71 \( 1 + (-9.48 - 5.47i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (11.0 + 6.35i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.97 + 8.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.23iT - 83T^{2} \)
89 \( 1 + (6.96 + 4.02i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.7 - 7.38i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.867445912027124091109652840944, −9.094347108565139397347152297399, −8.371661358041476032357895287977, −7.52702003494392048982483485656, −6.24456976319716464356768225269, −5.74338896097918376540161945169, −4.82279416162062358601326702521, −3.50275936378508489380042868022, −2.16710985873664932799802477921, −1.39273865984335507043950117415, 0.55052903990460384078834401036, 2.42775378709702930696539187061, 2.83667113960726365086676735987, 4.80216913286050819357694593698, 5.65564123415000155798834688243, 6.23505669364995519324636778009, 7.02438763142148081049042598940, 7.927567289123906740207807508085, 8.875494226913885785211010855977, 9.675365580373750367868613355067

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