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Properties

Label 1-28e2-784.27-r0-0-0
Degree $1$
Conductor $784$
Sign $0.999 + 0.00801i$
Analytic cond. $3.64088$
Root an. cond. $3.64088$
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  + (−0.433 + 0.900i)3-s + (−0.433 + 0.900i)5-s + (−0.623 − 0.781i)9-s + (−0.781 − 0.623i)11-s + (0.781 + 0.623i)13-s + (−0.623 − 0.781i)15-s + (0.222 − 0.974i)17-s − i·19-s + (−0.222 − 0.974i)23-s + (−0.623 − 0.781i)25-s + (0.974 − 0.222i)27-s + (0.974 + 0.222i)29-s + 31-s + (0.900 − 0.433i)33-s + (−0.974 − 0.222i)37-s + ⋯
L(s)  = 1  + (−0.433 + 0.900i)3-s + (−0.433 + 0.900i)5-s + (−0.623 − 0.781i)9-s + (−0.781 − 0.623i)11-s + (0.781 + 0.623i)13-s + (−0.623 − 0.781i)15-s + (0.222 − 0.974i)17-s − i·19-s + (−0.222 − 0.974i)23-s + (−0.623 − 0.781i)25-s + (0.974 − 0.222i)27-s + (0.974 + 0.222i)29-s + 31-s + (0.900 − 0.433i)33-s + (−0.974 − 0.222i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.999 + 0.00801i$
Analytic conductor: \(3.64088\)
Root analytic conductor: \(3.64088\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 784,\ (0:\ ),\ 0.999 + 0.00801i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8941516751 + 0.003583004297i\)
\(L(\frac12)\) \(\approx\) \(0.8941516751 + 0.003583004297i\)
\(L(1)\) \(\approx\) \(0.7907624669 + 0.1859244491i\)
\(L(1)\) \(\approx\) \(0.7907624669 + 0.1859244491i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.433 + 0.900i)T \)
5 \( 1 + (-0.433 + 0.900i)T \)
11 \( 1 + (-0.781 - 0.623i)T \)
13 \( 1 + (0.781 + 0.623i)T \)
17 \( 1 + (0.222 - 0.974i)T \)
19 \( 1 - iT \)
23 \( 1 + (-0.222 - 0.974i)T \)
29 \( 1 + (0.974 + 0.222i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.974 - 0.222i)T \)
41 \( 1 + (-0.900 - 0.433i)T \)
43 \( 1 + (0.433 + 0.900i)T \)
47 \( 1 + (0.623 - 0.781i)T \)
53 \( 1 + (0.974 - 0.222i)T \)
59 \( 1 + (-0.433 - 0.900i)T \)
61 \( 1 + (-0.974 - 0.222i)T \)
67 \( 1 + iT \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 - T \)
83 \( 1 + (0.781 - 0.623i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.68489670971679429748997192209, −21.37222218062941381814085202082, −20.667734691160917303221347739530, −19.83742804459679717153388775380, −19.117770793168640902049952337672, −18.27529369212671236417729835063, −17.4658040682278950440226966524, −16.86212565781989987098156891940, −15.828982465701544794046315463821, −15.25665005674899279452074652448, −13.82027830912327807867871797631, −13.25465887403998150049060528093, −12.30364516502853593313847091124, −12.05333886492393577431380662476, −10.787645775968905297474257600493, −10.062070916644349677185824415078, −8.60101006275487229858692774902, −8.07360965562680089694217602028, −7.35382181519074330906340331022, −6.079076941768586808863855104932, −5.46569327660040408621431839103, −4.44571088882535363172996633505, −3.26498399501013003154776761823, −1.86681887621553577561640984881, −1.03295437612504881974033842684, 0.53603076867263618904038006823, 2.588356946281102564722449105050, 3.285274248081766230715273337755, 4.31703331135869667655362724099, 5.17738069102335673198165387628, 6.28532048554658245891236682473, 6.94977274680870782336084536973, 8.235258559988973930921321728021, 9.006032821332726178994330166523, 10.16757904527813315764932676173, 10.72469790929329778012060238521, 11.459991624670723043323678808501, 12.136339567317425303668150482449, 13.62959885803675016841682583917, 14.175387950094303407418448599612, 15.259245562093504113713840111056, 15.86175123129585455624882563451, 16.37481674565050416573297638047, 17.542220999784504040343993022024, 18.32132840488198122914347224354, 18.96448030490726466556470318084, 20.023549938535393984288599524534, 20.9234700928099314673781866549, 21.54680659556728456479150164453, 22.308708467132119106840369594162

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