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Properties

Label 2-702-117.83-c1-0-13
Degree $2$
Conductor $702$
Sign $0.317 + 0.948i$
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  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (0.926 − 3.45i)5-s + (4.42 − 1.18i)7-s + (0.707 − 0.707i)8-s − 3.57i·10-s + (0.277 + 1.03i)11-s + (−3.47 + 0.956i)13-s + (3.96 − 2.29i)14-s + (0.500 − 0.866i)16-s − 4.13·17-s + (−3.55 + 3.55i)19-s + (−0.926 − 3.45i)20-s + (0.536 + 0.928i)22-s + (1.26 + 2.18i)23-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.433 − 0.249i)4-s + (0.414 − 1.54i)5-s + (1.67 − 0.448i)7-s + (0.249 − 0.249i)8-s − 1.13i·10-s + (0.0836 + 0.312i)11-s + (−0.964 + 0.265i)13-s + (1.06 − 0.612i)14-s + (0.125 − 0.216i)16-s − 1.00·17-s + (−0.814 + 0.814i)19-s + (−0.207 − 0.772i)20-s + (0.114 + 0.197i)22-s + (0.263 + 0.456i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $0.317 + 0.948i$
Analytic conductor: \(5.60549\)
Root analytic conductor: \(2.36759\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{702} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 702,\ (\ :1/2),\ 0.317 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18072 - 1.57029i\)
\(L(\frac12)\) \(\approx\) \(2.18072 - 1.57029i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 \)
13 \( 1 + (3.47 - 0.956i)T \)
good5 \( 1 + (-0.926 + 3.45i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-4.42 + 1.18i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.277 - 1.03i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + 4.13T + 17T^{2} \)
19 \( 1 + (3.55 - 3.55i)T - 19iT^{2} \)
23 \( 1 + (-1.26 - 2.18i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.654 - 0.377i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-8.68 - 2.32i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.53 - 1.53i)T + 37iT^{2} \)
41 \( 1 + (0.112 - 0.419i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.15 + 1.24i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.366 + 1.36i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 11.0iT - 53T^{2} \)
59 \( 1 + (9.66 + 2.58i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-5.37 + 9.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.82 - 2.36i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (2.42 + 2.42i)T + 71iT^{2} \)
73 \( 1 + (-3.17 - 3.17i)T + 73iT^{2} \)
79 \( 1 + (0.638 - 1.10i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.13 + 1.64i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (6.96 - 6.96i)T - 89iT^{2} \)
97 \( 1 + (-0.666 - 2.48i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.37964040982296871745751673789, −9.420509545420442554783363056251, −8.458756174148883351133833629612, −7.82157950368081445736611836155, −6.60817711403094358440856914073, −5.34299986821979403107957070266, −4.69630230058956844223116907302, −4.25798956292361137540752167057, −2.18150016625220709362957930440, −1.30751505199061061093089931271, 2.18671102448849937066007415918, 2.74561273062324700740286356888, 4.34035501850540542721544379086, 5.11618509637169620768378127093, 6.24613142028856740700986932037, 6.92112680232185620611332161374, 7.85761212791000608754576223034, 8.699575505565173007187302695048, 10.01922137179293002388972421976, 10.93711185976064191666638024355

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