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Properties

Label 2-702-117.103-c1-0-6
Degree $2$
Conductor $702$
Sign $0.970 + 0.241i$
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 + (0.419 + 0.242i)5-s + (4.37 − 2.52i)7-s + 0.999i·8-s − 0.484·10-s + (2.78 − 1.60i)11-s + (−0.722 + 3.53i)13-s + (−2.52 + 4.37i)14-s + (−0.5 − 0.866i)16-s − 4.20·17-s − 3.21i·19-s + (0.419 − 0.242i)20-s + (−1.60 + 2.78i)22-s + (3.13 − 5.43i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.187 + 0.108i)5-s + (1.65 − 0.955i)7-s + 0.353i·8-s − 0.153·10-s + (0.838 − 0.484i)11-s + (−0.200 + 0.979i)13-s + (−0.675 + 1.16i)14-s + (−0.125 − 0.216i)16-s − 1.01·17-s − 0.736i·19-s + (0.0937 − 0.0541i)20-s + (−0.342 + 0.592i)22-s + (0.653 − 1.13i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $0.970 + 0.241i$
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: \((2,\ 702,\ (\ :1/2),\ 0.970 + 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39268 - 0.170709i\)
\(L(\frac12)\) \(\approx\) \(1.39268 - 0.170709i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.722 - 3.53i)T \)
good5 \( 1 + (-0.419 - 0.242i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-4.37 + 2.52i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.78 + 1.60i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 4.20T + 17T^{2} \)
19 \( 1 + 3.21iT - 19T^{2} \)
23 \( 1 + (-3.13 + 5.43i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.29 - 3.97i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.61 + 3.24i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.08iT - 37T^{2} \)
41 \( 1 + (-9.57 - 5.52i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.73 - 8.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.57 + 2.64i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.41T + 53T^{2} \)
59 \( 1 + (3.13 + 1.81i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.500 - 0.867i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.936 - 0.540i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.63iT - 71T^{2} \)
73 \( 1 - 0.325iT - 73T^{2} \)
79 \( 1 + (-3.91 - 6.78i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.08 + 2.93i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 8.42iT - 89T^{2} \)
97 \( 1 + (-11.3 + 6.52i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.55115825042285957301219495649, −9.317669205473576912685329994291, −8.723665841064165268165160527011, −7.83113947902049636025934728816, −6.97490644492772586255266064848, −6.26055491307034449638073056835, −4.78215110319945908896926000317, −4.23900068862688850853186852712, −2.27243999679749392414225407402, −1.06049422694143909365559047265, 1.47047293553300652548019968994, 2.34704882453362132474654478047, 3.90782589901344218958458761530, 5.09583940665502503909231224207, 5.88972106993174284694283563866, 7.35609883921442050883427732217, 7.931378990562882695114888169486, 8.999952480688752053747129538529, 9.312170843855763427243031679413, 10.67004961405840192959325130697

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