Properties

Label 2-690-115.114-c2-0-6
Degree $2$
Conductor $690$
Sign $-0.867 + 0.496i$
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  + 1.41i·2-s + 1.73i·3-s − 2.00·4-s + (−0.120 − 4.99i)5-s − 2.44·6-s − 0.825·7-s − 2.82i·8-s − 2.99·9-s + (7.06 − 0.170i)10-s + 20.1i·11-s − 3.46i·12-s − 8.65i·13-s − 1.16i·14-s + (8.65 − 0.208i)15-s + 4.00·16-s + 10.6·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.500·4-s + (−0.0241 − 0.999i)5-s − 0.408·6-s − 0.117·7-s − 0.353i·8-s − 0.333·9-s + (0.706 − 0.0170i)10-s + 1.82i·11-s − 0.288i·12-s − 0.665i·13-s − 0.0834i·14-s + (0.577 − 0.0139i)15-s + 0.250·16-s + 0.626·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.496i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.867 + 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.867 + 0.496i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ -0.867 + 0.496i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4830924846\)
\(L(\frac12)\) \(\approx\) \(0.4830924846\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 - 1.73iT \)
5 \( 1 + (0.120 + 4.99i)T \)
23 \( 1 + (10.9 + 20.2i)T \)
good7 \( 1 + 0.825T + 49T^{2} \)
11 \( 1 - 20.1iT - 121T^{2} \)
13 \( 1 + 8.65iT - 169T^{2} \)
17 \( 1 - 10.6T + 289T^{2} \)
19 \( 1 - 23.3iT - 361T^{2} \)
29 \( 1 + 27.2T + 841T^{2} \)
31 \( 1 + 3.63T + 961T^{2} \)
37 \( 1 + 21.2T + 1.36e3T^{2} \)
41 \( 1 + 56.4T + 1.68e3T^{2} \)
43 \( 1 + 37.6T + 1.84e3T^{2} \)
47 \( 1 - 37.2iT - 2.20e3T^{2} \)
53 \( 1 + 19.9T + 2.80e3T^{2} \)
59 \( 1 - 20.2T + 3.48e3T^{2} \)
61 \( 1 - 36.3iT - 3.72e3T^{2} \)
67 \( 1 + 43.4T + 4.48e3T^{2} \)
71 \( 1 + 76.9T + 5.04e3T^{2} \)
73 \( 1 + 73.5iT - 5.32e3T^{2} \)
79 \( 1 - 74.2iT - 6.24e3T^{2} \)
83 \( 1 - 115.T + 6.88e3T^{2} \)
89 \( 1 + 85.7iT - 7.92e3T^{2} \)
97 \( 1 + 117.T + 9.40e3T^{2} \)
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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

−10.22849865943199208574465173938, −9.950113066953324677406434877324, −9.023285093711395931947165585774, −8.114053450583101600129881614383, −7.45667820900459644357694110752, −6.19973717693273119390581961600, −5.23556538186575133939841215900, −4.58770025674402638583531449735, −3.59660945293452341683622160301, −1.72091802308372803039932191695, 0.16783066214608974369543701877, 1.75347539373637794562155278392, 3.05269651599744303510288653344, 3.62579979007288294745960173755, 5.29730971502545485247346641630, 6.22591506845268141711834809414, 7.08706534401082173510135278504, 8.099809931294265049316134026911, 8.947748537446351809751450537703, 9.889867033585891766646748151630

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