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Properties

Label 2-693-7.4-c1-0-31
Degree $2$
Conductor $693$
Sign $-0.276 + 0.961i$
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  + (0.328 + 0.568i)2-s + (0.784 − 1.35i)4-s + (−1.78 − 3.09i)5-s + (1.78 − 1.95i)7-s + 2.34·8-s + (1.17 − 2.02i)10-s + (−0.5 + 0.866i)11-s − 5.91·13-s + (1.69 + 0.373i)14-s + (−0.799 − 1.38i)16-s + (−0.828 + 1.43i)17-s + (−0.740 − 1.28i)19-s − 5.59·20-s − 0.656·22-s + (1.67 + 2.89i)23-s + ⋯
L(s)  = 1  + (0.232 + 0.402i)2-s + (0.392 − 0.679i)4-s + (−0.798 − 1.38i)5-s + (0.674 − 0.738i)7-s + 0.828·8-s + (0.370 − 0.641i)10-s + (−0.150 + 0.261i)11-s − 1.63·13-s + (0.453 + 0.0997i)14-s + (−0.199 − 0.346i)16-s + (−0.200 + 0.347i)17-s + (−0.169 − 0.294i)19-s − 1.25·20-s − 0.139·22-s + (0.348 + 0.603i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.276 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.276 + 0.961i$
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: \((2,\ 693,\ (\ :1/2),\ -0.276 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.874878 - 1.16189i\)
\(L(\frac12)\) \(\approx\) \(0.874878 - 1.16189i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.78 + 1.95i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.328 - 0.568i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.78 + 3.09i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 5.91T + 13T^{2} \)
17 \( 1 + (0.828 - 1.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.740 + 1.28i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.67 - 2.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.08T + 29T^{2} \)
31 \( 1 + (-3.54 + 6.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.25 - 3.90i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.28T + 41T^{2} \)
43 \( 1 - 1.59T + 43T^{2} \)
47 \( 1 + (0.828 + 1.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.61 + 7.98i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.42 + 7.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.34 - 5.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.91 + 8.50i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.61T + 71T^{2} \)
73 \( 1 + (2.28 - 3.95i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.19 - 5.53i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.167T + 83T^{2} \)
89 \( 1 + (1.28 + 2.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.73T + 97T^{2} \)
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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

−10.11280787137689447111261518629, −9.436853606453094193560421071538, −8.189797496874319416349876218613, −7.61396711686626525383726713013, −6.82965450038549147707673384652, −5.36982388711976944605973871500, −4.82204155482329666228820784666, −4.10689318302230619976533362654, −2.05821715277273023202709714564, −0.69277665132456341794769097435, 2.36094971793377371762365416199, 2.86179139187953556390210927162, 4.04800180638069445903359698884, 5.08380877763248858249569337224, 6.52395861108278404280855387082, 7.37197197862302662085256315877, 7.84009993969900803932144689602, 8.887070867993963575953854473330, 10.22537007291694541398880356365, 10.89618534709501754323476139874

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