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Properties

Label 2-1274-13.10-c1-0-11
Degree $2$
Conductor $1274$
Sign $-0.105 - 0.994i$
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 + (1.14 + 1.98i)3-s + (0.499 − 0.866i)4-s + 0.901i·5-s + (1.98 + 1.14i)6-s − 0.999i·8-s + (−1.12 + 1.94i)9-s + (0.450 + 0.781i)10-s + (−3.75 + 2.16i)11-s + 2.29·12-s + (0.426 + 3.58i)13-s + (−1.78 + 1.03i)15-s + (−0.5 − 0.866i)16-s + (−2.53 + 4.38i)17-s + 2.24i·18-s + (−5.34 − 3.08i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.661 + 1.14i)3-s + (0.249 − 0.433i)4-s + 0.403i·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 + 0.661·12-s + (0.118 + 0.992i)13-s + (−0.461 + 0.266i)15-s + (−0.125 − 0.216i)16-s + (−0.614 + 1.06i)17-s + 0.529i·18-s + (−1.22 − 0.708i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $-0.105 - 0.994i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ -0.105 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.513817214\)
\(L(\frac12)\) \(\approx\) \(2.513817214\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.426 - 3.58i)T \)
good3 \( 1 + (-1.14 - 1.98i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 0.901iT - 5T^{2} \)
11 \( 1 + (3.75 - 2.16i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.53 - 4.38i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.34 + 3.08i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.22 - 7.31i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.09 - 1.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.873iT - 31T^{2} \)
37 \( 1 + (-0.124 + 0.0721i)T + (18.5 - 32.0i)T^{2} \)
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 \( 1 + 2.92iT - 47T^{2} \)
53 \( 1 - 1.69T + 53T^{2} \)
59 \( 1 + (-7.40 - 4.27i)T + (29.5 + 51.0i)T^{2} \)
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 \( 1 - 0.539iT - 73T^{2} \)
79 \( 1 - 6.53T + 79T^{2} \)
83 \( 1 + 13.2iT - 83T^{2} \)
89 \( 1 + (-6.74 + 3.89i)T + (44.5 - 77.0i)T^{2} \)
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

−10.06212583445869375949205249708, −9.114247742135192965286757635986, −8.630231128620160130602207284157, −7.31670302780054410484137020614, −6.58251025636398364688532275381, −5.35068009328151938646002292425, −4.53803681955544990657277931469, −3.89317827492025007267966306713, −2.90390452179639051374479232363, −2.01012426260802480805343353803, 0.76672800611808317118856092130, 2.47716090404466220260025683717, 2.89542611340453277802556707857, 4.41467704877505627542831876945, 5.26833192422054246087460722828, 6.26111012904433053898397166985, 6.97253537078389849457447607124, 7.962705509612502145811796351632, 8.263198364325406139262757964694, 9.052921559357616520451257911136

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