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Properties

Label 2-690-115.114-c2-0-37
Degree $2$
Conductor $690$
Sign $-0.269 + 0.962i$
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 + (1.71 − 4.69i)5-s + 2.44·6-s + 12.9·7-s + 2.82i·8-s − 2.99·9-s + (−6.63 − 2.43i)10-s − 18.3i·11-s − 3.46i·12-s − 7.60i·13-s − 18.2i·14-s + (8.13 + 2.97i)15-s + 4.00·16-s − 19.9·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.500·4-s + (0.343 − 0.939i)5-s + 0.408·6-s + 1.84·7-s + 0.353i·8-s − 0.333·9-s + (−0.663 − 0.243i)10-s − 1.67i·11-s − 0.288i·12-s − 0.584i·13-s − 1.30i·14-s + (0.542 + 0.198i)15-s + 0.250·16-s − 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.269 + 0.962i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.269 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.269 + 0.962i$
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: \((2,\ 690,\ (\ :1),\ -0.269 + 0.962i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.075141204\)
\(L(\frac12)\) \(\approx\) \(2.075141204\)
\(L(2)\) 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 \( 1 + (-1.71 + 4.69i)T \)
23 \( 1 + (-22.9 + 1.79i)T \)
good7 \( 1 - 12.9T + 49T^{2} \)
11 \( 1 + 18.3iT - 121T^{2} \)
13 \( 1 + 7.60iT - 169T^{2} \)
17 \( 1 + 19.9T + 289T^{2} \)
19 \( 1 - 24.3iT - 361T^{2} \)
29 \( 1 - 32.2T + 841T^{2} \)
31 \( 1 - 3.18T + 961T^{2} \)
37 \( 1 + 11.0T + 1.36e3T^{2} \)
41 \( 1 + 60.4T + 1.68e3T^{2} \)
43 \( 1 - 50.3T + 1.84e3T^{2} \)
47 \( 1 + 79.4iT - 2.20e3T^{2} \)
53 \( 1 + 92.1T + 2.80e3T^{2} \)
59 \( 1 - 5.16T + 3.48e3T^{2} \)
61 \( 1 + 10.5iT - 3.72e3T^{2} \)
67 \( 1 + 72.2T + 4.48e3T^{2} \)
71 \( 1 + 8.94T + 5.04e3T^{2} \)
73 \( 1 + 70.6iT - 5.32e3T^{2} \)
79 \( 1 + 1.35iT - 6.24e3T^{2} \)
83 \( 1 - 135.T + 6.88e3T^{2} \)
89 \( 1 + 41.9iT - 7.92e3T^{2} \)
97 \( 1 - 165.T + 9.40e3T^{2} \)
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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

−10.28572097749379183689354487425, −8.929910307196298764445987065869, −8.561732442680813342110373115093, −7.934308212118586912806002763896, −6.03005049907760323863458450132, −5.13628183248678458653840018079, −4.58679506303183713351897374556, −3.39605063778187865466223480589, −1.92468135872232324881799994531, −0.78803717288121158104398441063, 1.59018702488713827290120204246, 2.53184311074283321694385107171, 4.54267769201689899895046250225, 4.92006335120485820012238349724, 6.39538089179470789769766961740, 7.08606821250173425044700244266, 7.59262830365449133055960573286, 8.655810850777100802939681870331, 9.445124321864800201297577641509, 10.68328656973144776035076639572

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