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Properties

Label 2-693-7.4-c1-0-7
Degree $2$
Conductor $693$
Sign $-0.934 + 0.354i$
Analytic cond. $5.53363$
Root an. cond. $2.35236$
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  + (1.39 + 2.40i)2-s + (−2.86 + 4.96i)4-s + (0.412 + 0.715i)5-s + (2.63 + 0.257i)7-s − 10.3·8-s + (−1.14 + 1.98i)10-s + (−0.5 + 0.866i)11-s − 0.296·13-s + (3.04 + 6.70i)14-s + (−8.72 − 15.1i)16-s + (−3.34 + 5.79i)17-s + (1.41 + 2.45i)19-s − 4.73·20-s − 2.78·22-s + (−1.98 − 3.43i)23-s + ⋯
L(s)  = 1  + (0.983 + 1.70i)2-s + (−1.43 + 2.48i)4-s + (0.184 + 0.319i)5-s + (0.995 + 0.0971i)7-s − 3.67·8-s + (−0.363 + 0.629i)10-s + (−0.150 + 0.261i)11-s − 0.0823·13-s + (0.813 + 1.79i)14-s + (−2.18 − 3.77i)16-s + (−0.811 + 1.40i)17-s + (0.325 + 0.562i)19-s − 1.05·20-s − 0.593·22-s + (−0.413 − 0.716i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.934 + 0.354i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ -0.934 + 0.354i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.408783 - 2.22805i\)
\(L(\frac12)\) \(\approx\) \(0.408783 - 2.22805i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-2.63 - 0.257i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-1.39 - 2.40i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.412 - 0.715i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 0.296T + 13T^{2} \)
17 \( 1 + (3.34 - 5.79i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.41 - 2.45i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.98 + 3.43i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.484T + 29T^{2} \)
31 \( 1 + (-3.66 + 6.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.86 - 4.96i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.645T + 41T^{2} \)
43 \( 1 - 6.43T + 43T^{2} \)
47 \( 1 + (-3.86 - 6.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.55 + 6.16i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.578 - 1.00i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.63 + 4.56i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.50 - 2.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.58T + 71T^{2} \)
73 \( 1 + (8.01 - 13.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.16 - 3.74i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.37T + 83T^{2} \)
89 \( 1 + (-6.08 - 10.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.7T + 97T^{2} \)
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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

−11.11390394757734275769967758730, −9.896160185657778554299653381094, −8.588822972782976926706951653211, −8.185367181857123867884902036279, −7.32860099601726574321884091443, −6.33803480799238588702481836031, −5.77371767210510929930891657209, −4.60639343145168684996990703334, −4.08827073955024393522000824196, −2.54335474407397683004359898698, 0.920240972329514785461180459295, 2.14157014486405603342395612558, 3.15325825105246043716377920074, 4.43353051433278972174826423443, 5.00257846002199437642452315961, 5.80751172000778127262183021381, 7.28050172727516334823927558483, 8.837964587790223466385416977495, 9.268510863271673956782228430145, 10.37640468014159833524882417491

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