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Properties

Label 2-28e2-112.11-c0-0-0
Degree $2$
Conductor $784$
Sign $0.262 - 0.964i$
Analytic cond. $0.391266$
Root an. cond. $0.625513$
Motivic weight $0$
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 + (0.499 − 0.866i)4-s + 0.999i·8-s + (−0.866 + 0.5i)9-s + (0.366 + 1.36i)11-s + (−0.5 − 0.866i)16-s + (0.499 − 0.866i)18-s + (−1 − 0.999i)22-s + (1 + 1.73i)23-s + (0.866 + 0.5i)25-s + (1 − i)29-s + (0.866 + 0.499i)32-s + 0.999i·36-s + (0.366 − 1.36i)37-s + (−1 + i)43-s + (1.36 + 0.366i)44-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + 0.999i·8-s + (−0.866 + 0.5i)9-s + (0.366 + 1.36i)11-s + (−0.5 − 0.866i)16-s + (0.499 − 0.866i)18-s + (−1 − 0.999i)22-s + (1 + 1.73i)23-s + (0.866 + 0.5i)25-s + (1 − i)29-s + (0.866 + 0.499i)32-s + 0.999i·36-s + (0.366 − 1.36i)37-s + (−1 + i)43-s + (1.36 + 0.366i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.262 - 0.964i$
Analytic conductor: \(0.391266\)
Root analytic conductor: \(0.625513\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :0),\ 0.262 - 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6173234465\)
\(L(\frac12)\) \(\approx\) \(0.6173234465\)
\(L(1)\) 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 \( 1 \)
good3 \( 1 + (0.866 - 0.5i)T^{2} \)
5 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-1 + i)T - iT^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (1 - i)T - iT^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.866 + 0.5i)T^{2} \)
67 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−10.57029073188762336181744799289, −9.562977689745740868936565662641, −9.121287237823490297336067979194, −8.024377452603596106401427103273, −7.38702074153223646808147123751, −6.51118404834811116607209632008, −5.48065549015786640702895597504, −4.64931098517031085029855821421, −2.90683863404783294727796390998, −1.63348733417979984724315325363, 0.921235182822194093828361976923, 2.76289769754977387620640774126, 3.39027847338787516607996878872, 4.84760465597166231463613346532, 6.31352074638419705772509222319, 6.75625300580935746008734521081, 8.300592192779580659484683779682, 8.559353326531623616510025640126, 9.326700697669372271014373808969, 10.51508315443421977841203956729

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