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Properties

Label 2-28e2-28.23-c0-0-0
Degree $2$
Conductor $784$
Sign $-0.198 - 0.980i$
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  + (−0.707 + 1.22i)5-s + (−0.5 + 0.866i)9-s − 1.41·13-s + (0.707 + 1.22i)17-s + (−0.499 − 0.866i)25-s + 1.41·41-s + (−0.707 − 1.22i)45-s + (1 + 1.73i)53-s + (0.707 − 1.22i)61-s + (1.00 − 1.73i)65-s + (−0.707 − 1.22i)73-s + (−0.499 − 0.866i)81-s − 2·85-s + (0.707 − 1.22i)89-s + 1.41·97-s + ⋯
L(s)  = 1  + (−0.707 + 1.22i)5-s + (−0.5 + 0.866i)9-s − 1.41·13-s + (0.707 + 1.22i)17-s + (−0.499 − 0.866i)25-s + 1.41·41-s + (−0.707 − 1.22i)45-s + (1 + 1.73i)53-s + (0.707 − 1.22i)61-s + (1.00 − 1.73i)65-s + (−0.707 − 1.22i)73-s + (−0.499 − 0.866i)81-s − 2·85-s + (0.707 − 1.22i)89-s + 1.41·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.198 - 0.980i$
Analytic conductor: \(0.391266\)
Root analytic conductor: \(0.625513\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :0),\ -0.198 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7031534483\)
\(L(\frac12)\) \(\approx\) \(0.7031534483\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + 1.41T + T^{2} \)
17 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - 1.41T + T^{2} \)
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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

−10.61593492303972501322188148698, −10.23089915609164333846559559247, −9.027188267707737621998027614473, −7.75744778669207605121185528137, −7.60986422251963171409731073248, −6.45996446998392831085189214008, −5.46471344791170937108420421516, −4.31672083984664065764507750897, −3.19098863271732626725752172074, −2.27145856328926302083055343475, 0.73058791624488996240477705339, 2.67287580141634008099689034067, 3.91441462222660223569178048558, 4.88987706741787007671724744373, 5.60030813630381783338285901441, 6.96712928911483035470699059094, 7.72969510672359168174373962945, 8.632520942195642668044530656237, 9.357287367129295314758539129777, 10.00815912151496021496313700394

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