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Properties

Label 2-690-5.2-c2-0-26
Degree $2$
Conductor $690$
Sign $0.908 + 0.417i$
Analytic cond. $18.8011$
Root an. cond. $4.33602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)2-s + (−1.22 + 1.22i)3-s + 2i·4-s + (−4.90 − 0.988i)5-s − 2.44·6-s + (2.14 + 2.14i)7-s + (−2 + 2i)8-s − 2.99i·9-s + (−3.91 − 5.89i)10-s − 12.9·11-s + (−2.44 − 2.44i)12-s + (10.2 − 10.2i)13-s + 4.29i·14-s + (7.21 − 4.79i)15-s − 4·16-s + (−2.98 − 2.98i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s + (−0.980 − 0.197i)5-s − 0.408·6-s + (0.307 + 0.307i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (−0.391 − 0.589i)10-s − 1.18·11-s + (−0.204 − 0.204i)12-s + (0.788 − 0.788i)13-s + 0.307i·14-s + (0.480 − 0.319i)15-s − 0.250·16-s + (−0.175 − 0.175i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.908 + 0.417i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.908 + 0.417i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.181742001\)
\(L(\frac12)\) \(\approx\) \(1.181742001\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 + (4.90 + 0.988i)T \)
23 \( 1 + (-3.39 + 3.39i)T \)
good7 \( 1 + (-2.14 - 2.14i)T + 49iT^{2} \)
11 \( 1 + 12.9T + 121T^{2} \)
13 \( 1 + (-10.2 + 10.2i)T - 169iT^{2} \)
17 \( 1 + (2.98 + 2.98i)T + 289iT^{2} \)
19 \( 1 + 26.4iT - 361T^{2} \)
29 \( 1 - 45.6iT - 841T^{2} \)
31 \( 1 - 37.4T + 961T^{2} \)
37 \( 1 + (44.9 + 44.9i)T + 1.36e3iT^{2} \)
41 \( 1 - 39.3T + 1.68e3T^{2} \)
43 \( 1 + (-29.7 + 29.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (10.2 + 10.2i)T + 2.20e3iT^{2} \)
53 \( 1 + (-34.6 + 34.6i)T - 2.80e3iT^{2} \)
59 \( 1 + 22.8iT - 3.48e3T^{2} \)
61 \( 1 + 70.3T + 3.72e3T^{2} \)
67 \( 1 + (-19.6 - 19.6i)T + 4.48e3iT^{2} \)
71 \( 1 - 131.T + 5.04e3T^{2} \)
73 \( 1 + (-49.0 + 49.0i)T - 5.32e3iT^{2} \)
79 \( 1 + 59.2iT - 6.24e3T^{2} \)
83 \( 1 + (31.0 - 31.0i)T - 6.88e3iT^{2} \)
89 \( 1 + 169. iT - 7.92e3T^{2} \)
97 \( 1 + (-8.17 - 8.17i)T + 9.40e3iT^{2} \)
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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

−10.60469520383933469517849673670, −9.048081628763746263368557476088, −8.410779457350918817629942001148, −7.53973989256002761419325140128, −6.66719833632297643338239295752, −5.33914522339570580116389434783, −4.97524307264127032425251980930, −3.80357819651536314130525685039, −2.78153096761927388357036821660, −0.43392932404901623490299174244, 1.15707646936634119575460002253, 2.60554862120441367480029342155, 3.88225654624906871989308355013, 4.62734551214964500488800647413, 5.81131891706583080783204511108, 6.67803540909435447086084707302, 7.82624311415203701430602044901, 8.274427657654910192474273181825, 9.751771808648753134758281058379, 10.72139686836353864621455633694

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