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Properties

Label 2-702-117.59-c1-0-11
Degree $2$
Conductor $702$
Sign $-0.911 + 0.411i$
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 + (−0.763 − 2.84i)5-s + (−0.504 − 1.88i)7-s + (−0.707 − 0.707i)8-s + (−2.55 − 1.47i)10-s + (0.566 + 0.566i)11-s + (2.11 + 2.92i)13-s + (−1.68 − 0.975i)14-s − 1.00·16-s + (−4.02 − 6.96i)17-s + (−1.18 + 4.41i)19-s + (−2.84 + 0.763i)20-s + 0.801·22-s + (−3.49 − 6.05i)23-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.341 − 1.27i)5-s + (−0.190 − 0.712i)7-s + (−0.250 − 0.250i)8-s + (−0.807 − 0.466i)10-s + (0.170 + 0.170i)11-s + (0.586 + 0.810i)13-s + (−0.451 − 0.260i)14-s − 0.250·16-s + (−0.975 − 1.68i)17-s + (−0.271 + 1.01i)19-s + (−0.636 + 0.170i)20-s + 0.170·22-s + (−0.729 − 1.26i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 + 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $-0.911 + 0.411i$
Analytic conductor: \(5.60549\)
Root analytic conductor: \(2.36759\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{702} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 702,\ (\ :1/2),\ -0.911 + 0.411i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.312006 - 1.44948i\)
\(L(\frac12)\) \(\approx\) \(0.312006 - 1.44948i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.11 - 2.92i)T \)
good5 \( 1 + (0.763 + 2.84i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.504 + 1.88i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.566 - 0.566i)T + 11iT^{2} \)
17 \( 1 + (4.02 + 6.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.18 - 4.41i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.49 + 6.05i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.87iT - 29T^{2} \)
31 \( 1 + (2.31 - 0.619i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.09 + 7.82i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-5.00 - 1.34i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.02 - 1.16i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.37 + 5.14i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + 8.27iT - 53T^{2} \)
59 \( 1 + (-6.24 - 6.24i)T + 59iT^{2} \)
61 \( 1 + (-2.46 + 4.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.200 + 0.749i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.57 + 0.689i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-5.68 + 5.68i)T - 73iT^{2} \)
79 \( 1 + (-0.554 - 0.960i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.29 + 0.614i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (2.84 - 0.761i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.54 - 0.950i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.17552700215661574266864543358, −9.118009812073816701737836300810, −8.664353160353434651287582347911, −7.35811881178869698683198484663, −6.47937895622312363590943317105, −5.22096084823259519932881736251, −4.40754283202033560432450327711, −3.74488285846645690264559047622, −2.06001918265972764501027605562, −0.65268407058164866416452620063, 2.34616328420422296056870928937, 3.37766407016509604540658075293, 4.25477197085253448295848947416, 5.83052603983922803182083043064, 6.20192087231190344784837829181, 7.20381982391519468610149937877, 8.097025612277040497655497838867, 8.882621310015432963414435564980, 10.07393447464332087879616439604, 11.01874279300840587947863996067

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