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Properties

Label 2-28e2-112.53-c1-0-8
Degree $2$
Conductor $784$
Sign $-0.996 + 0.0794i$
Analytic cond. $6.26027$
Root an. cond. $2.50205$
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  + (−1.08 − 0.909i)2-s + (0.672 + 2.51i)3-s + (0.343 + 1.97i)4-s + (−0.780 + 2.91i)5-s + (1.55 − 3.33i)6-s + (1.42 − 2.44i)8-s + (−3.25 + 1.87i)9-s + (3.49 − 2.44i)10-s + (−3.12 + 0.838i)11-s + (−4.71 + 2.18i)12-s + (−2.52 + 2.52i)13-s − 7.83·15-s + (−3.76 + 1.35i)16-s + (−0.201 + 0.348i)17-s + (5.23 + 0.927i)18-s + (−1.39 − 0.373i)19-s + ⋯
L(s)  = 1  + (−0.765 − 0.643i)2-s + (0.388 + 1.44i)3-s + (0.171 + 0.985i)4-s + (−0.348 + 1.30i)5-s + (0.635 − 1.35i)6-s + (0.502 − 0.864i)8-s + (−1.08 + 0.626i)9-s + (1.10 − 0.772i)10-s + (−0.943 + 0.252i)11-s + (−1.36 + 0.631i)12-s + (−0.699 + 0.699i)13-s − 2.02·15-s + (−0.940 + 0.338i)16-s + (−0.0487 + 0.0844i)17-s + (1.23 + 0.218i)18-s + (−0.319 − 0.0855i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.996 + 0.0794i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.996 + 0.0794i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0272783 - 0.685272i\)
\(L(\frac12)\) \(\approx\) \(0.0272783 - 0.685272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.08 + 0.909i)T \)
7 \( 1 \)
good3 \( 1 + (-0.672 - 2.51i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (0.780 - 2.91i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (3.12 - 0.838i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (2.52 - 2.52i)T - 13iT^{2} \)
17 \( 1 + (0.201 - 0.348i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.39 + 0.373i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-7.89 + 4.55i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.47 + 1.47i)T - 29iT^{2} \)
31 \( 1 + (2.12 - 3.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.520 - 1.94i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 8.96iT - 41T^{2} \)
43 \( 1 + (0.997 + 0.997i)T + 43iT^{2} \)
47 \( 1 + (-2.09 - 3.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.82 - 0.488i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.37 - 0.636i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.55 + 0.685i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-3.12 - 11.6i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 0.451iT - 71T^{2} \)
73 \( 1 + (9.40 + 5.43i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.31 - 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.742 - 0.742i)T - 83iT^{2} \)
89 \( 1 + (-11.1 + 6.41i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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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

−10.70295049744176522097514413797, −10.05448955369716376277663640968, −9.219871273772691940491396417897, −8.510696130589498278344809837968, −7.39411189604850881312648949557, −6.76181037868216290993700433713, −4.98309436464663191825250570808, −4.09948201467047867755017730917, −3.07969794545922599617268393798, −2.47688403800748055930378009004, 0.42515159691346040417090919424, 1.48012992819899381773974566655, 2.78911171733624157743312476068, 4.84820150768400755304239922242, 5.52217907642714568407973095879, 6.66296407976945640148023627694, 7.65504443114894077002895058637, 7.908248656550513525317391148104, 8.747024996056030168975243167614, 9.443477970873349953185329377547

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