Properties

Label 2-690-115.114-c2-0-36
Degree $2$
Conductor $690$
Sign $-0.614 + 0.788i$
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 + (4.76 − 1.52i)5-s + 2.44·6-s − 5.00·7-s + 2.82i·8-s − 2.99·9-s + (−2.16 − 6.73i)10-s + 0.0576i·11-s − 3.46i·12-s − 4.70i·13-s + 7.08i·14-s + (2.64 + 8.24i)15-s + 4.00·16-s − 12.7·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.500·4-s + (0.952 − 0.305i)5-s + 0.408·6-s − 0.715·7-s + 0.353i·8-s − 0.333·9-s + (−0.216 − 0.673i)10-s + 0.00524i·11-s − 0.288i·12-s − 0.362i·13-s + 0.506i·14-s + (0.176 + 0.549i)15-s + 0.250·16-s − 0.749·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.614 + 0.788i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.614 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.614 + 0.788i$
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: \((2,\ 690,\ (\ :1),\ -0.614 + 0.788i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.271201890\)
\(L(\frac12)\) \(\approx\) \(1.271201890\)
\(L(2)\) 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 \( 1 + (-4.76 + 1.52i)T \)
23 \( 1 + (-19.0 + 12.9i)T \)
good7 \( 1 + 5.00T + 49T^{2} \)
11 \( 1 - 0.0576iT - 121T^{2} \)
13 \( 1 + 4.70iT - 169T^{2} \)
17 \( 1 + 12.7T + 289T^{2} \)
19 \( 1 + 27.4iT - 361T^{2} \)
29 \( 1 + 49.2T + 841T^{2} \)
31 \( 1 + 19.1T + 961T^{2} \)
37 \( 1 - 27.0T + 1.36e3T^{2} \)
41 \( 1 - 37.5T + 1.68e3T^{2} \)
43 \( 1 - 42.1T + 1.84e3T^{2} \)
47 \( 1 + 89.2iT - 2.20e3T^{2} \)
53 \( 1 - 90.8T + 2.80e3T^{2} \)
59 \( 1 + 84.9T + 3.48e3T^{2} \)
61 \( 1 + 108. iT - 3.72e3T^{2} \)
67 \( 1 + 59.1T + 4.48e3T^{2} \)
71 \( 1 + 124.T + 5.04e3T^{2} \)
73 \( 1 + 73.2iT - 5.32e3T^{2} \)
79 \( 1 - 82.2iT - 6.24e3T^{2} \)
83 \( 1 + 16.6T + 6.88e3T^{2} \)
89 \( 1 + 129. iT - 7.92e3T^{2} \)
97 \( 1 + 96.0T + 9.40e3T^{2} \)
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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

−9.956203074354953483991825412748, −9.115833855189354264792437536649, −8.894870068135155236286388733167, −7.28889703995738506343869996856, −6.17552104246757835753651091837, −5.26406117415110180768091004219, −4.36123540327468069044191869927, −3.12024716400771596178435444368, −2.17643510937532580081758905230, −0.44727872305821303695449044013, 1.50624442609220657713880331799, 2.84738390771656670962420515080, 4.16163835733613907588938037218, 5.71056963254324364146964366131, 6.01582762544375405311885151676, 7.05678019417636149760694733124, 7.66278023298073549678444656054, 9.044520123869289419163765538216, 9.385535020491864936170311373474, 10.45134307181785344570694893264

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