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Properties

Label 2-1274-13.4-c1-0-42
Degree $2$
Conductor $1274$
Sign $-0.938 + 0.344i$
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 + (1.24 − 2.16i)3-s + (0.499 + 0.866i)4-s − 2.78i·5-s + (−2.16 + 1.24i)6-s − 0.999i·8-s + (−1.61 − 2.79i)9-s + (−1.39 + 2.41i)10-s + (3.26 + 1.88i)11-s + 2.49·12-s + (3.40 − 1.19i)13-s + (−6.02 − 3.47i)15-s + (−0.5 + 0.866i)16-s + (−2.50 − 4.34i)17-s + 3.22i·18-s + (2.68 − 1.54i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.720 − 1.24i)3-s + (0.249 + 0.433i)4-s − 1.24i·5-s + (−0.882 + 0.509i)6-s − 0.353i·8-s + (−0.538 − 0.932i)9-s + (−0.440 + 0.763i)10-s + (0.984 + 0.568i)11-s + 0.720·12-s + (0.943 − 0.331i)13-s + (−1.55 − 0.897i)15-s + (−0.125 + 0.216i)16-s + (−0.607 − 1.05i)17-s + 0.761i·18-s + (0.614 − 0.355i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.344i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.938 + 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $-0.938 + 0.344i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ -0.938 + 0.344i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.662986340\)
\(L(\frac12)\) \(\approx\) \(1.662986340\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.40 + 1.19i)T \)
good3 \( 1 + (-1.24 + 2.16i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 2.78iT - 5T^{2} \)
11 \( 1 + (-3.26 - 1.88i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.50 + 4.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.68 + 1.54i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0487 + 0.0843i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.34 - 2.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.24iT - 31T^{2} \)
37 \( 1 + (-0.424 - 0.244i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (8.29 + 4.79i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.642 - 1.11i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.86iT - 47T^{2} \)
53 \( 1 + 10.2T + 53T^{2} \)
59 \( 1 + (4.70 - 2.71i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.31 - 2.27i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.32 - 4.22i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.07 - 2.92i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 15.3iT - 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 - 16.0iT - 83T^{2} \)
89 \( 1 + (-12.4 - 7.16i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.50 + 3.18i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.148163541422294073039339595112, −8.580276558326329926516567346018, −7.84122576712399412466187011095, −7.10838423536933510846946411728, −6.30593436883049496773344447773, −5.01178792262399253972320073408, −3.90525812465561444882136786180, −2.67130664993453861465222417216, −1.57171218104208027569937527033, −0.855326074848206757252877822359, 1.76910217890211652656463168833, 3.33323193148390330656319224176, 3.59797553513985069999429981780, 4.86089884406739349855339294855, 6.27115800872477833492566599458, 6.57398664483239262522361795309, 7.80555399392228042308293744572, 8.685283863737396804654248874258, 9.098124361514771426459606462929, 10.03405528552624323393416790554

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