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Properties

Label 2-28e2-112.59-c1-0-67
Degree $2$
Conductor $784$
Sign $-0.725 + 0.687i$
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  + (0.757 + 1.19i)2-s + (−0.817 − 3.05i)3-s + (−0.851 + 1.80i)4-s + (0.797 + 0.213i)5-s + (3.02 − 3.28i)6-s + (−2.80 + 0.353i)8-s + (−6.05 + 3.49i)9-s + (0.349 + 1.11i)10-s + (−0.732 − 2.73i)11-s + (6.22 + 1.12i)12-s + (−2.91 − 2.91i)13-s − 2.61i·15-s + (−2.54 − 3.08i)16-s + (−2.30 − 1.33i)17-s + (−8.75 − 4.57i)18-s + (0.436 + 0.117i)19-s + ⋯
L(s)  = 1  + (0.535 + 0.844i)2-s + (−0.472 − 1.76i)3-s + (−0.425 + 0.904i)4-s + (0.356 + 0.0956i)5-s + (1.23 − 1.34i)6-s + (−0.992 + 0.125i)8-s + (−2.01 + 1.16i)9-s + (0.110 + 0.352i)10-s + (−0.220 − 0.823i)11-s + (1.79 + 0.323i)12-s + (−0.807 − 0.807i)13-s − 0.673i·15-s + (−0.637 − 0.770i)16-s + (−0.558 − 0.322i)17-s + (−2.06 − 1.07i)18-s + (0.100 + 0.0268i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.725 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.725 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.725 + 0.687i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.725 + 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.272272 - 0.683293i\)
\(L(\frac12)\) \(\approx\) \(0.272272 - 0.683293i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.757 - 1.19i)T \)
7 \( 1 \)
good3 \( 1 + (0.817 + 3.05i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (-0.797 - 0.213i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.732 + 2.73i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (2.91 + 2.91i)T + 13iT^{2} \)
17 \( 1 + (2.30 + 1.33i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.436 - 0.117i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.63 - 2.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.04 - 2.04i)T - 29iT^{2} \)
31 \( 1 + (-1.26 + 2.18i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.625 + 2.33i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 11.9T + 41T^{2} \)
43 \( 1 + (3.27 - 3.27i)T - 43iT^{2} \)
47 \( 1 + (4.98 + 8.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.24 + 0.869i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.66 + 1.51i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.65 + 6.18i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-5.97 + 1.60i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 5.14T + 71T^{2} \)
73 \( 1 + (3.49 - 6.05i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.72 - 5.61i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.39 + 5.39i)T - 83iT^{2} \)
89 \( 1 + (0.528 + 0.915i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.2iT - 97T^{2} \)
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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

−9.893837754602866941882744911062, −8.571292192155519916391081804787, −7.998285282005206860883255620420, −7.16544433118048717105494279166, −6.56032251157809215722211592536, −5.66035954236764711997959600701, −5.12769192676316787587273273707, −3.27678644258644470183891351216, −2.17501708447494291367496481997, −0.30384422903019000765024183292, 2.12527415568764995133882685355, 3.39010969352119713031620139862, 4.43624966975460904168567067701, 4.85311839513819026369464379582, 5.74597503826653696751455503044, 6.79555020049039509426880423873, 8.613856912432413464392527279059, 9.411603319747126775679701559606, 9.965435484160823972899228980897, 10.45207806180522738670963041796

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