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Properties

Label 2-1690-13.4-c1-0-3
Degree $2$
Conductor $1690$
Sign $-0.711 + 0.702i$
Analytic cond. $13.4947$
Root an. cond. $3.67351$
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  + (0.866 + 0.5i)2-s + (−0.366 + 0.633i)3-s + (0.499 + 0.866i)4-s + i·5-s + (−0.633 + 0.366i)6-s + (−2.59 + 1.5i)7-s + 0.999i·8-s + (1.23 + 2.13i)9-s + (−0.5 + 0.866i)10-s + (−2.59 − 1.5i)11-s − 0.732·12-s − 3·14-s + (−0.633 − 0.366i)15-s + (−0.5 + 0.866i)16-s + (−4.09 − 7.09i)17-s + 2.46i·18-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.211 + 0.366i)3-s + (0.249 + 0.433i)4-s + 0.447i·5-s + (−0.258 + 0.149i)6-s + (−0.981 + 0.566i)7-s + 0.353i·8-s + (0.410 + 0.711i)9-s + (−0.158 + 0.273i)10-s + (−0.783 − 0.452i)11-s − 0.211·12-s − 0.801·14-s + (−0.163 − 0.0945i)15-s + (−0.125 + 0.216i)16-s + (−0.993 − 1.72i)17-s + 0.580i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1690\)    =    \(2 \cdot 5 \cdot 13^{2}\)
Sign: $-0.711 + 0.702i$
Analytic conductor: \(13.4947\)
Root analytic conductor: \(3.67351\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1690} (1161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1690,\ (\ :1/2),\ -0.711 + 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5973969668\)
\(L(\frac12)\) \(\approx\) \(0.5973969668\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 - iT \)
13 \( 1 \)
good3 \( 1 + (0.366 - 0.633i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (2.59 - 1.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (4.09 + 7.09i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.401 + 0.232i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.73 - 8.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.26 + 2.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.73iT - 31T^{2} \)
37 \( 1 + (-0.696 - 0.401i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-9 - 5.19i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 - 0.464T + 53T^{2} \)
59 \( 1 + (9 - 5.19i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.09 + 5.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.19 + 3i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 11.6iT - 73T^{2} \)
79 \( 1 + 4.19T + 79T^{2} \)
83 \( 1 - 8.19iT - 83T^{2} \)
89 \( 1 + (5.89 + 3.40i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.90 - 4.56i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.658286222128799683268421444737, −9.322523363175444521005118463927, −7.935727092441941425068120520698, −7.45516236965581991571357155186, −6.47282071809513532850166004374, −5.74514745990836504979174473998, −5.03666706566587814907778903519, −4.08421821567275985356975623442, −3.01090587160604180286135945189, −2.32162378615252066444111136693, 0.17569834611816404218082576969, 1.57987442111881741198968819990, 2.75339008262501818742218440716, 3.98608395646959901098930521370, 4.38353007987954978905711890612, 5.68812772276304152422557023025, 6.43361644048266595437449235010, 6.92824501927833694461136179712, 8.027122754327057325306731588568, 8.916684014343956674437498109604

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