Properties

Label 2-690-5.4-c1-0-10
Degree $2$
Conductor $690$
Sign $-0.139 - 0.990i$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
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  + i·2-s + i·3-s − 4-s + (2.21 − 0.311i)5-s − 6-s + 0.622i·7-s i·8-s − 9-s + (0.311 + 2.21i)10-s + 4.42·11-s i·12-s − 1.37i·13-s − 0.622·14-s + (0.311 + 2.21i)15-s + 16-s + 6.42i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.990 − 0.139i)5-s − 0.408·6-s + 0.235i·7-s − 0.353i·8-s − 0.333·9-s + (0.0983 + 0.700i)10-s + 1.33·11-s − 0.288i·12-s − 0.382i·13-s − 0.166·14-s + (0.0803 + 0.571i)15-s + 0.250·16-s + 1.55i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.139 - 0.990i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.139 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16191 + 1.33657i\)
\(L(\frac12)\) \(\approx\) \(1.16191 + 1.33657i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 - iT \)
5 \( 1 + (-2.21 + 0.311i)T \)
23 \( 1 + iT \)
good7 \( 1 - 0.622iT - 7T^{2} \)
11 \( 1 - 4.42T + 11T^{2} \)
13 \( 1 + 1.37iT - 13T^{2} \)
17 \( 1 - 6.42iT - 17T^{2} \)
19 \( 1 - 1.37T + 19T^{2} \)
29 \( 1 - 4.23T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 11.8iT - 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 - 1.05iT - 43T^{2} \)
47 \( 1 + 11.6iT - 47T^{2} \)
53 \( 1 + 3.37iT - 53T^{2} \)
59 \( 1 + 9.61T + 59T^{2} \)
61 \( 1 - 8.66T + 61T^{2} \)
67 \( 1 - 11.8iT - 67T^{2} \)
71 \( 1 + 6.99T + 71T^{2} \)
73 \( 1 + 2.75iT - 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 + 2.19iT - 83T^{2} \)
89 \( 1 + 2.75T + 89T^{2} \)
97 \( 1 + 2.13iT - 97T^{2} \)
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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

−10.22371823113874228432298043510, −9.951442228148650106749997038885, −8.713377022360766856081001092503, −8.531516920192260308199570271724, −6.92979940637423393874711620708, −6.19631737320808570744148549817, −5.44541610131936594297340118509, −4.42056191506382165798247338144, −3.31249983328288677068321483242, −1.57560809319713457757397663122, 1.10442929055303463146516145853, 2.20330752512858384749545525974, 3.36444271991513187527168419979, 4.67200353072705918438445387483, 5.75226381288091066140001981433, 6.72178235877510302429754953997, 7.47196905249769108096788146089, 9.007347732895186106275361086333, 9.249311416817895034016428195055, 10.23419076093685965307146247809

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