Properties

Label 2-690-3.2-c2-0-25
Degree $2$
Conductor $690$
Sign $-0.847 - 0.531i$
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 + (−1.59 + 2.54i)3-s − 2.00·4-s + 2.23i·5-s + (−3.59 − 2.25i)6-s + 10.3·7-s − 2.82i·8-s + (−3.92 − 8.10i)9-s − 3.16·10-s + 14.4i·11-s + (3.18 − 5.08i)12-s + 14.5·13-s + 14.5i·14-s + (−5.68 − 3.56i)15-s + 4.00·16-s − 0.438i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.531 + 0.847i)3-s − 0.500·4-s + 0.447i·5-s + (−0.599 − 0.375i)6-s + 1.47·7-s − 0.353i·8-s + (−0.435 − 0.900i)9-s − 0.316·10-s + 1.31i·11-s + (0.265 − 0.423i)12-s + 1.12·13-s + 1.04i·14-s + (−0.378 − 0.237i)15-s + 0.250·16-s − 0.0257i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 - 0.531i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.847 - 0.531i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ -0.847 - 0.531i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.792124478\)
\(L(\frac12)\) \(\approx\) \(1.792124478\)
\(L(2)\) 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 \( 1 + (1.59 - 2.54i)T \)
5 \( 1 - 2.23iT \)
23 \( 1 - 4.79iT \)
good7 \( 1 - 10.3T + 49T^{2} \)
11 \( 1 - 14.4iT - 121T^{2} \)
13 \( 1 - 14.5T + 169T^{2} \)
17 \( 1 + 0.438iT - 289T^{2} \)
19 \( 1 - 30.5T + 361T^{2} \)
29 \( 1 - 23.6iT - 841T^{2} \)
31 \( 1 - 37.8T + 961T^{2} \)
37 \( 1 - 3.95T + 1.36e3T^{2} \)
41 \( 1 - 0.897iT - 1.68e3T^{2} \)
43 \( 1 + 47.5T + 1.84e3T^{2} \)
47 \( 1 - 39.9iT - 2.20e3T^{2} \)
53 \( 1 + 67.0iT - 2.80e3T^{2} \)
59 \( 1 + 43.7iT - 3.48e3T^{2} \)
61 \( 1 + 69.1T + 3.72e3T^{2} \)
67 \( 1 - 15.9T + 4.48e3T^{2} \)
71 \( 1 + 32.7iT - 5.04e3T^{2} \)
73 \( 1 + 136.T + 5.32e3T^{2} \)
79 \( 1 + 117.T + 6.24e3T^{2} \)
83 \( 1 + 72.3iT - 6.88e3T^{2} \)
89 \( 1 - 48.7iT - 7.92e3T^{2} \)
97 \( 1 - 128.T + 9.40e3T^{2} \)
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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

−10.51770018917795748072523007883, −9.808622610639547791350419520510, −8.873890229163253143158504999958, −7.949627685727703295206141872659, −7.10667768309471226534686876400, −6.07446308911607043619693946076, −5.06951367354411170527030383186, −4.54913812728969085740720495907, −3.34514085138136966657764883527, −1.39029547423532614776486635058, 0.846146480351116671107964335104, 1.52115990028203470654015271232, 2.99719281704540928740028301815, 4.41938810315252604306979724778, 5.39997453081499335314333070601, 6.06732450401704904441998495345, 7.52847832409368008103111618806, 8.310862652723513540310781804363, 8.763171199405511290173670029691, 10.20981925719390202031335381558

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