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Properties

Label 2-702-117.32-c1-0-8
Degree $2$
Conductor $702$
Sign $0.720 + 0.693i$
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 + (2.64 − 0.707i)5-s + (3.36 − 0.902i)7-s + (0.707 − 0.707i)8-s + (−2.36 − 1.36i)10-s + (1.29 − 1.29i)11-s + (2.77 + 2.29i)13-s + (−3.01 − 1.74i)14-s − 1.00·16-s + (0.419 + 0.726i)17-s + (−6.94 − 1.86i)19-s + (0.707 + 2.64i)20-s − 1.82·22-s + (3.18 + 5.52i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (1.18 − 0.316i)5-s + (1.27 − 0.340i)7-s + (0.250 − 0.250i)8-s + (−0.748 − 0.432i)10-s + (0.389 − 0.389i)11-s + (0.770 + 0.637i)13-s + (−0.806 − 0.465i)14-s − 0.250·16-s + (0.101 + 0.176i)17-s + (−1.59 − 0.427i)19-s + (0.158 + 0.590i)20-s − 0.389·22-s + (0.665 + 1.15i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $0.720 + 0.693i$
Analytic conductor: \(5.60549\)
Root analytic conductor: \(2.36759\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{702} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 702,\ (\ :1/2),\ 0.720 + 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56212 - 0.629826i\)
\(L(\frac12)\) \(\approx\) \(1.56212 - 0.629826i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.77 - 2.29i)T \)
good5 \( 1 + (-2.64 + 0.707i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-3.36 + 0.902i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.29 + 1.29i)T - 11iT^{2} \)
17 \( 1 + (-0.419 - 0.726i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.94 + 1.86i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.18 - 5.52i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.05iT - 29T^{2} \)
31 \( 1 + (1.99 + 7.43i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (4.09 - 1.09i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.07 + 4.01i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-7.47 - 4.31i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.46 + 2.00i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + 3.92iT - 53T^{2} \)
59 \( 1 + (-5.26 + 5.26i)T - 59iT^{2} \)
61 \( 1 + (-0.247 + 0.428i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.34 - 1.43i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.48 - 5.55i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (9.66 + 9.66i)T + 73iT^{2} \)
79 \( 1 + (0.892 + 1.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.07 + 15.2i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (0.385 + 1.43i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.50 - 16.7i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.45936143150338775656224038777, −9.331147867925560732225786025895, −8.852954033344638957831731694054, −8.005606631325938241268372829421, −6.85822214125170866516921436308, −5.85426627161652447526811946212, −4.80697941145460039433301780402, −3.74155083253890917186127000890, −2.06673224650260400720672808099, −1.35385179176965790915060349729, 1.47058505310247295218186695625, 2.45211709201660339443541662416, 4.32910760224077796982068064791, 5.38160920288164943261958959679, 6.14702208241355784572851749656, 6.95025209473239280944314852408, 8.213987577513626398539247332480, 8.644898516824455762856102224329, 9.640715830678897172474216369095, 10.58445176048088926321564296190

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