Properties

Label 2-28e2-196.115-c1-0-0
Degree $2$
Conductor $784$
Sign $-0.279 - 0.960i$
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  + (−1.00 − 0.309i)3-s + (−2.49 − 2.69i)5-s + (−1.72 − 2.00i)7-s + (−1.56 − 1.06i)9-s + (1.80 + 2.64i)11-s + (0.664 − 1.38i)13-s + (1.67 + 3.47i)15-s + (−4.82 + 1.89i)17-s + (0.131 + 0.228i)19-s + (1.10 + 2.54i)21-s + (4.84 + 1.90i)23-s + (−0.633 + 8.45i)25-s + (3.20 + 4.02i)27-s + (3.81 − 4.78i)29-s + (−2.31 + 4.01i)31-s + ⋯
L(s)  = 1  + (−0.579 − 0.178i)3-s + (−1.11 − 1.20i)5-s + (−0.650 − 0.759i)7-s + (−0.522 − 0.356i)9-s + (0.543 + 0.796i)11-s + (0.184 − 0.382i)13-s + (0.431 + 0.896i)15-s + (−1.16 + 0.458i)17-s + (0.0302 + 0.0523i)19-s + (0.240 + 0.556i)21-s + (1.00 + 0.396i)23-s + (−0.126 + 1.69i)25-s + (0.616 + 0.773i)27-s + (0.708 − 0.888i)29-s + (−0.416 + 0.720i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.279 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.279 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.279 - 0.960i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.279 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0297803 + 0.0396897i\)
\(L(\frac12)\) \(\approx\) \(0.0297803 + 0.0396897i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (1.72 + 2.00i)T \)
good3 \( 1 + (1.00 + 0.309i)T + (2.47 + 1.68i)T^{2} \)
5 \( 1 + (2.49 + 2.69i)T + (-0.373 + 4.98i)T^{2} \)
11 \( 1 + (-1.80 - 2.64i)T + (-4.01 + 10.2i)T^{2} \)
13 \( 1 + (-0.664 + 1.38i)T + (-8.10 - 10.1i)T^{2} \)
17 \( 1 + (4.82 - 1.89i)T + (12.4 - 11.5i)T^{2} \)
19 \( 1 + (-0.131 - 0.228i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.84 - 1.90i)T + (16.8 + 15.6i)T^{2} \)
29 \( 1 + (-3.81 + 4.78i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (2.31 - 4.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.04 - 0.158i)T + (35.3 + 10.9i)T^{2} \)
41 \( 1 + (10.7 - 2.44i)T + (36.9 - 17.7i)T^{2} \)
43 \( 1 + (-3.75 - 0.856i)T + (38.7 + 18.6i)T^{2} \)
47 \( 1 + (-0.532 - 7.10i)T + (-46.4 + 7.00i)T^{2} \)
53 \( 1 + (0.650 - 0.0979i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (7.61 + 7.06i)T + (4.40 + 58.8i)T^{2} \)
61 \( 1 + (-1.80 + 11.9i)T + (-58.2 - 17.9i)T^{2} \)
67 \( 1 + (3.37 + 1.94i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.90 - 6.30i)T + (15.7 - 69.2i)T^{2} \)
73 \( 1 + (-1.99 - 0.149i)T + (72.1 + 10.8i)T^{2} \)
79 \( 1 + (10.5 - 6.10i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.11 + 3.90i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (2.98 - 4.38i)T + (-32.5 - 82.8i)T^{2} \)
97 \( 1 + 7.53iT - 97T^{2} \)
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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

−10.74904078986866639386444065201, −9.556143267038184567206530330260, −8.842745885443757273518963977099, −8.009044947906946852526612650169, −7.00904436683219056882643212778, −6.29984984199007729135239458411, −5.00196700152854684606356024959, −4.28444185326138507587655840487, −3.31653731154018275115836358762, −1.18419584051593288307686139296, 0.03036514950067334886090654944, 2.62447981717344759522862054244, 3.38695881170149313522601988821, 4.54961825776354768673199141069, 5.76506611604401085035474334367, 6.60163271690385998748076554992, 7.17923671404275116890133209323, 8.534095991397204931818301450951, 8.980440466015504834016355368575, 10.38457479888007989150437043565

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