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Properties

Label 2-702-39.8-c1-0-2
Degree $2$
Conductor $702$
Sign $-0.819 - 0.572i$
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 + (0.730 − 0.730i)5-s + (−2.49 + 2.49i)7-s + (0.707 + 0.707i)8-s + 1.03i·10-s + (1.95 + 1.95i)11-s + (−3.21 + 1.63i)13-s − 3.53i·14-s − 1.00·16-s + 5.44·17-s + (−5.99 − 5.99i)19-s + (−0.730 − 0.730i)20-s − 2.76·22-s − 5.11·23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.326 − 0.326i)5-s + (−0.943 + 0.943i)7-s + (0.250 + 0.250i)8-s + 0.326i·10-s + (0.589 + 0.589i)11-s + (−0.891 + 0.452i)13-s − 0.943i·14-s − 0.250·16-s + 1.31·17-s + (−1.37 − 1.37i)19-s + (−0.163 − 0.163i)20-s − 0.589·22-s − 1.06·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 - 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.819 - 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $-0.819 - 0.572i$
Analytic conductor: \(5.60549\)
Root analytic conductor: \(2.36759\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{702} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 702,\ (\ :1/2),\ -0.819 - 0.572i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.201530 + 0.640605i\)
\(L(\frac12)\) \(\approx\) \(0.201530 + 0.640605i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.21 - 1.63i)T \)
good5 \( 1 + (-0.730 + 0.730i)T - 5iT^{2} \)
7 \( 1 + (2.49 - 2.49i)T - 7iT^{2} \)
11 \( 1 + (-1.95 - 1.95i)T + 11iT^{2} \)
17 \( 1 - 5.44T + 17T^{2} \)
19 \( 1 + (5.99 + 5.99i)T + 19iT^{2} \)
23 \( 1 + 5.11T + 23T^{2} \)
29 \( 1 - 8.51iT - 29T^{2} \)
31 \( 1 + (-3.53 - 3.53i)T + 31iT^{2} \)
37 \( 1 + (7.78 - 7.78i)T - 37iT^{2} \)
41 \( 1 + (-1.06 + 1.06i)T - 41iT^{2} \)
43 \( 1 - 3.79iT - 43T^{2} \)
47 \( 1 + (7.80 + 7.80i)T + 47iT^{2} \)
53 \( 1 - 11.7iT - 53T^{2} \)
59 \( 1 + (1.86 + 1.86i)T + 59iT^{2} \)
61 \( 1 - 6.42T + 61T^{2} \)
67 \( 1 + (-0.887 - 0.887i)T + 67iT^{2} \)
71 \( 1 + (3.17 - 3.17i)T - 71iT^{2} \)
73 \( 1 + (2.31 - 2.31i)T - 73iT^{2} \)
79 \( 1 + 5.29T + 79T^{2} \)
83 \( 1 + (-4.77 + 4.77i)T - 83iT^{2} \)
89 \( 1 + (-0.144 - 0.144i)T + 89iT^{2} \)
97 \( 1 + (4.66 + 4.66i)T + 97iT^{2} \)
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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

−10.41567300298297046939205318506, −9.722286699747034430433611657456, −9.099183224406339934862309538993, −8.427440127121715629825102643984, −7.09753538487408095097912358144, −6.55806460208659242353415488798, −5.52483657531177137560321471219, −4.65697170805242917012034971939, −3.05929711693807899491072444414, −1.74328055215103137468226215348, 0.39940818260940694118374727059, 2.12750052903670536579336871964, 3.43130773221357005137772229426, 4.13637090068740132769861066329, 5.87247869420813903894750911229, 6.53825639819973943256680686388, 7.68394003319765276328635622690, 8.301645827922021875983412019173, 9.691713911421878759454607216076, 10.05439110920678122207159774985

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