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Properties

Label 2-28e2-16.5-c1-0-66
Degree $2$
Conductor $784$
Sign $-0.281 + 0.959i$
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  + (1.29 − 0.576i)2-s + (0.715 − 0.715i)3-s + (1.33 − 1.48i)4-s + (−0.867 − 0.867i)5-s + (0.511 − 1.33i)6-s + (0.867 − 2.69i)8-s + 1.97i·9-s + (−1.62 − 0.620i)10-s + (−2.97 − 2.97i)11-s + (−0.109 − 2.02i)12-s + (2.02 − 2.02i)13-s − 1.24·15-s + (−0.430 − 3.97i)16-s − 0.264·17-s + (1.13 + 2.55i)18-s + (4.53 − 4.53i)19-s + ⋯
L(s)  = 1  + (0.913 − 0.407i)2-s + (0.412 − 0.412i)3-s + (0.667 − 0.744i)4-s + (−0.388 − 0.388i)5-s + (0.208 − 0.545i)6-s + (0.306 − 0.951i)8-s + 0.658i·9-s + (−0.512 − 0.196i)10-s + (−0.897 − 0.897i)11-s + (−0.0314 − 0.583i)12-s + (0.560 − 0.560i)13-s − 0.320·15-s + (−0.107 − 0.994i)16-s − 0.0641·17-s + (0.268 + 0.601i)18-s + (1.04 − 1.04i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.281 + 0.959i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.281 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64950 - 2.20176i\)
\(L(\frac12)\) \(\approx\) \(1.64950 - 2.20176i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 + 0.576i)T \)
7 \( 1 \)
good3 \( 1 + (-0.715 + 0.715i)T - 3iT^{2} \)
5 \( 1 + (0.867 + 0.867i)T + 5iT^{2} \)
11 \( 1 + (2.97 + 2.97i)T + 11iT^{2} \)
13 \( 1 + (-2.02 + 2.02i)T - 13iT^{2} \)
17 \( 1 + 0.264T + 17T^{2} \)
19 \( 1 + (-4.53 + 4.53i)T - 19iT^{2} \)
23 \( 1 - 1.54iT - 23T^{2} \)
29 \( 1 + (0.328 - 0.328i)T - 29iT^{2} \)
31 \( 1 + 6.04T + 31T^{2} \)
37 \( 1 + (-6.64 - 6.64i)T + 37iT^{2} \)
41 \( 1 - 11.0iT - 41T^{2} \)
43 \( 1 + (-3.38 - 3.38i)T + 43iT^{2} \)
47 \( 1 + 3.12T + 47T^{2} \)
53 \( 1 + (-0.430 - 0.430i)T + 53iT^{2} \)
59 \( 1 + (-4.62 - 4.62i)T + 59iT^{2} \)
61 \( 1 + (4.86 - 4.86i)T - 61iT^{2} \)
67 \( 1 + (-3.34 + 3.34i)T - 67iT^{2} \)
71 \( 1 - 9.03iT - 71T^{2} \)
73 \( 1 + 14.8iT - 73T^{2} \)
79 \( 1 - 12.5T + 79T^{2} \)
83 \( 1 + (-0.715 + 0.715i)T - 83iT^{2} \)
89 \( 1 - 10.9iT - 89T^{2} \)
97 \( 1 - 14.2T + 97T^{2} \)
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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

−10.32389934964899345110487617512, −9.205192741017804485085028198065, −8.088996872321787864948241073850, −7.59560487954385763668296233105, −6.36479273392860806429608133703, −5.37713787151725996468794017306, −4.66549334272813125359157882240, −3.31048523031782415418093182010, −2.59077102078301249532613849489, −1.03772484785817523874299042843, 2.17166827901120509499756931725, 3.45570688750590340611159246979, 3.97098000723446281397920249155, 5.14556478496955645786350836133, 6.06293934109629357930022979361, 7.17077439801816304923955166525, 7.66625868639836817960371754982, 8.765562118085852703245389460662, 9.670413935785727264359341988885, 10.67754552079877398316049458891

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