Properties

Label 2-690-115.114-c2-0-9
Degree $2$
Conductor $690$
Sign $0.890 - 0.454i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.890 - 0.454i$
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: \((2,\ 690,\ (\ :1),\ 0.890 - 0.454i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9396140236\)
\(L(\frac12)\) \(\approx\) \(0.9396140236\)
\(L(2)\) 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

−10.18734462106224019895728788262, −9.517573082845895684167956766626, −8.767426314361510322691367288787, −7.77957666367816721991221755590, −6.97317921104521011786516384374, −5.63737166286500278414418925010, −4.70718772506420479545134291325, −3.85845545485814965042185742016, −2.10010790706377198872184364512, −1.46817817803781578690053790276, 0.33725178281746386328809221969, 2.76211478944952459494903665246, 3.58269499178911058751132112773, 4.83585014353003624616014527825, 5.86208803633495191879989699182, 6.52366016668693745517041565801, 7.55941048811361425541008815061, 8.510659765610163530802287467010, 9.155899385863529694940268708441, 10.27087949160053957262522339042

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