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Properties

Label 2-702-39.5-c1-0-3
Degree $2$
Conductor $702$
Sign $0.0865 - 0.996i$
Analytic cond. $5.60549$
Root an. cond. $2.36759$
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  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.136 − 0.136i)5-s + (3.40 + 3.40i)7-s + (0.707 − 0.707i)8-s + 0.193i·10-s + (−2.95 + 2.95i)11-s + (−3.31 + 1.41i)13-s − 4.82i·14-s − 1.00·16-s + 1.26·17-s + (−2.82 + 2.82i)19-s + (0.136 − 0.136i)20-s + 4.17·22-s − 7.83·23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.0610 − 0.0610i)5-s + (1.28 + 1.28i)7-s + (0.250 − 0.250i)8-s + 0.0610i·10-s + (−0.889 + 0.889i)11-s + (−0.919 + 0.393i)13-s − 1.28i·14-s − 0.250·16-s + 0.306·17-s + (−0.647 + 0.647i)19-s + (0.0305 − 0.0305i)20-s + 0.889·22-s − 1.63·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0865 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0865 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $0.0865 - 0.996i$
Analytic conductor: \(5.60549\)
Root analytic conductor: \(2.36759\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{702} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 702,\ (\ :1/2),\ 0.0865 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.651129 + 0.596995i\)
\(L(\frac12)\) \(\approx\) \(0.651129 + 0.596995i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
13 \( 1 + (3.31 - 1.41i)T \)
good5 \( 1 + (0.136 + 0.136i)T + 5iT^{2} \)
7 \( 1 + (-3.40 - 3.40i)T + 7iT^{2} \)
11 \( 1 + (2.95 - 2.95i)T - 11iT^{2} \)
17 \( 1 - 1.26T + 17T^{2} \)
19 \( 1 + (2.82 - 2.82i)T - 19iT^{2} \)
23 \( 1 + 7.83T + 23T^{2} \)
29 \( 1 + 2.30iT - 29T^{2} \)
31 \( 1 + (3.60 - 3.60i)T - 31iT^{2} \)
37 \( 1 + (-3.54 - 3.54i)T + 37iT^{2} \)
41 \( 1 + (-1.65 - 1.65i)T + 41iT^{2} \)
43 \( 1 - 4.36iT - 43T^{2} \)
47 \( 1 + (-5.75 + 5.75i)T - 47iT^{2} \)
53 \( 1 - 8.16iT - 53T^{2} \)
59 \( 1 + (7.77 - 7.77i)T - 59iT^{2} \)
61 \( 1 - 6.62T + 61T^{2} \)
67 \( 1 + (-8.52 + 8.52i)T - 67iT^{2} \)
71 \( 1 + (-4.58 - 4.58i)T + 71iT^{2} \)
73 \( 1 + (6.49 + 6.49i)T + 73iT^{2} \)
79 \( 1 - 2.03T + 79T^{2} \)
83 \( 1 + (-12.1 - 12.1i)T + 83iT^{2} \)
89 \( 1 + (-12.0 + 12.0i)T - 89iT^{2} \)
97 \( 1 + (-5.85 + 5.85i)T - 97iT^{2} \)
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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

−10.52510477207762428315992276112, −9.879262074246001905570303997842, −8.929421927709814133355046595178, −8.035914691902197255614400770802, −7.64863074415134189975423659414, −6.13961389551224442335088784435, −5.06891972856050424018566583818, −4.28251315618433470121400037813, −2.47597111379831889494074263759, −1.92703695089540771794477318040, 0.52344554303230457875350026857, 2.13581646189583132753234772835, 3.82624069443215798158961700060, 4.92178225083234635972753996545, 5.70514417392555110506857507186, 7.03324318100897431423267842319, 7.79725302271796327732887014378, 8.141045659726203014427251333295, 9.355250798267242316984577781445, 10.38904349383196503473205229001

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