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Properties

Label 2-702-117.2-c1-0-7
Degree $2$
Conductor $702$
Sign $0.899 - 0.436i$
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 + (−0.398 + 1.48i)5-s + (0.370 − 1.38i)7-s + (0.707 − 0.707i)8-s + (1.33 − 0.769i)10-s + (0.350 − 0.350i)11-s + (0.183 + 3.60i)13-s + (−1.23 + 0.715i)14-s − 1.00·16-s + (1.10 − 1.91i)17-s + (0.805 + 3.00i)19-s + (−1.48 − 0.398i)20-s − 0.496·22-s + (0.887 − 1.53i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.178 + 0.664i)5-s + (0.139 − 0.522i)7-s + (0.250 − 0.250i)8-s + (0.421 − 0.243i)10-s + (0.105 − 0.105i)11-s + (0.0508 + 0.998i)13-s + (−0.331 + 0.191i)14-s − 0.250·16-s + (0.268 − 0.464i)17-s + (0.184 + 0.689i)19-s + (−0.332 − 0.0890i)20-s − 0.105·22-s + (0.185 − 0.320i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $0.899 - 0.436i$
Analytic conductor: \(5.60549\)
Root analytic conductor: \(2.36759\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{702} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 702,\ (\ :1/2),\ 0.899 - 0.436i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06796 + 0.245194i\)
\(L(\frac12)\) \(\approx\) \(1.06796 + 0.245194i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.183 - 3.60i)T \)
good5 \( 1 + (0.398 - 1.48i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.370 + 1.38i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.350 + 0.350i)T - 11iT^{2} \)
17 \( 1 + (-1.10 + 1.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.805 - 3.00i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.887 + 1.53i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.71iT - 29T^{2} \)
31 \( 1 + (-0.303 - 0.0814i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.95 - 11.0i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-7.77 + 2.08i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.40 + 0.811i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.17 - 11.8i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 2.02iT - 53T^{2} \)
59 \( 1 + (-1.98 + 1.98i)T - 59iT^{2} \)
61 \( 1 + (3.39 + 5.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.05 + 7.65i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-10.8 + 2.90i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-4.45 - 4.45i)T + 73iT^{2} \)
79 \( 1 + (-4.77 + 8.26i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.13 - 2.17i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-3.41 - 0.914i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (7.11 + 1.90i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.66309237737542835506341655419, −9.685307613937098429624546668082, −8.954585914796355155645004298146, −7.889486328853895504122060105429, −7.13935708046386442960615348847, −6.33349062340807618068543111559, −4.84085375696467877561587547774, −3.77514443829349509152746913455, −2.78429537597504082080598323095, −1.31547826488006000871422468751, 0.78573434366397420869464605554, 2.42836310682583088348694883373, 3.99460046771410785356545328565, 5.21852196229154013908866941212, 5.80080812145178324396905728996, 7.02290934190235708942691495165, 7.908403352933434591345492629766, 8.643415451553706094180244342910, 9.301750633712909074905591406895, 10.26678407345839829004275799878

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