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Properties

Label 2-702-117.103-c1-0-10
Degree $2$
Conductor $702$
Sign $0.162 + 0.986i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.548 − 0.316i)5-s + (−2.15 + 1.24i)7-s − 0.999i·8-s − 0.633·10-s + (4.20 − 2.43i)11-s + (−0.541 − 3.56i)13-s + (−1.24 + 2.15i)14-s + (−0.5 − 0.866i)16-s + 6.27·17-s − 4.86i·19-s + (−0.548 + 0.316i)20-s + (2.43 − 4.20i)22-s + (1.77 − 3.07i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.245 − 0.141i)5-s + (−0.814 + 0.469i)7-s − 0.353i·8-s − 0.200·10-s + (1.26 − 0.732i)11-s + (−0.150 − 0.988i)13-s + (−0.332 + 0.575i)14-s + (−0.125 − 0.216i)16-s + 1.52·17-s − 1.11i·19-s + (−0.122 + 0.0707i)20-s + (0.518 − 0.897i)22-s + (0.369 − 0.640i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.162 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $0.162 + 0.986i$
Analytic conductor: \(5.60549\)
Root analytic conductor: \(2.36759\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{702} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 702,\ (\ :1/2),\ 0.162 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47623 - 1.25291i\)
\(L(\frac12)\) \(\approx\) \(1.47623 - 1.25291i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
13 \( 1 + (0.541 + 3.56i)T \)
good5 \( 1 + (0.548 + 0.316i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.15 - 1.24i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.20 + 2.43i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 6.27T + 17T^{2} \)
19 \( 1 + 4.86iT - 19T^{2} \)
23 \( 1 + (-1.77 + 3.07i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.415 + 0.719i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.73 + 2.15i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.81iT - 37T^{2} \)
41 \( 1 + (0.0678 + 0.0391i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.84 - 8.38i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.30 - 2.48i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 6.36T + 53T^{2} \)
59 \( 1 + (7.86 + 4.54i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.28 - 9.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.82 + 3.94i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.3iT - 71T^{2} \)
73 \( 1 - 1.05iT - 73T^{2} \)
79 \( 1 + (1.68 + 2.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-13.1 + 7.60i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 0.595iT - 89T^{2} \)
97 \( 1 + (-14.7 + 8.53i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.26331402986951499253431146845, −9.515531328515645836578670183629, −8.655431965633763642705592385059, −7.58395547533545642482443238251, −6.39181992999191738383081342554, −5.83073187077393233607323500043, −4.67913358365108459611498157334, −3.49940218418819399643182140858, −2.79962736618382972204038907440, −0.906675024855119968353494345099, 1.70554765704910145424692881547, 3.58996390089658055176987467963, 3.85544842734360900968435894677, 5.27398570778197961899356119485, 6.25153863588606405221994895063, 7.13468926968142168914653287255, 7.61494964483184945760017327750, 9.078298235585259124266197562395, 9.665645486997423547127614464289, 10.64670372975041778476599068400

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