↑

Properties

Label 2-10-5.2-c12-0-1
Degree $2$
Conductor $10$
Sign $-0.465 - 0.884i$
Analytic cond. $9.13993$
Root an. cond. $3.02323$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (32 + 32i)2-s + (203. − 203. i)3-s + 2.04e3i·4-s + (−1.17e4 + 1.02e4i)5-s + 1.30e4·6-s + (6.85e4 + 6.85e4i)7-s + (−6.55e4 + 6.55e4i)8-s + 4.48e5i·9-s + (−7.05e5 − 4.87e4i)10-s − 7.85e5·11-s + (4.16e5 + 4.16e5i)12-s + (−3.66e6 + 3.66e6i)13-s + 4.38e6i·14-s + (−3.09e5 + 4.48e6i)15-s − 4.19e6·16-s + (1.73e7 + 1.73e7i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (0.279 − 0.279i)3-s + 0.5i·4-s + (−0.754 + 0.656i)5-s + 0.279·6-s + (0.582 + 0.582i)7-s + (−0.250 + 0.250i)8-s + 0.844i·9-s + (−0.705 − 0.0487i)10-s − 0.443·11-s + (0.139 + 0.139i)12-s + (−0.760 + 0.760i)13-s + 0.582i·14-s + (−0.0272 + 0.393i)15-s − 0.250·16-s + (0.720 + 0.720i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.465 - 0.884i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.465 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.465 - 0.884i$
Analytic conductor: \(9.13993\)
Root analytic conductor: \(3.02323\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :6),\ -0.465 - 0.884i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.993794 + 1.64626i\)
\(L(\frac12)\) \(\approx\) \(0.993794 + 1.64626i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-32 - 32i)T \)
5 \( 1 + (1.17e4 - 1.02e4i)T \)
good3 \( 1 + (-203. + 203. i)T - 5.31e5iT^{2} \)
7 \( 1 + (-6.85e4 - 6.85e4i)T + 1.38e10iT^{2} \)
11 \( 1 + 7.85e5T + 3.13e12T^{2} \)
13 \( 1 + (3.66e6 - 3.66e6i)T - 2.32e13iT^{2} \)
17 \( 1 + (-1.73e7 - 1.73e7i)T + 5.82e14iT^{2} \)
19 \( 1 + 4.23e7iT - 2.21e15T^{2} \)
23 \( 1 + (-1.95e8 + 1.95e8i)T - 2.19e16iT^{2} \)
29 \( 1 - 5.16e8iT - 3.53e17T^{2} \)
31 \( 1 - 3.32e8T + 7.87e17T^{2} \)
37 \( 1 + (1.95e7 + 1.95e7i)T + 6.58e18iT^{2} \)
41 \( 1 - 2.51e9T + 2.25e19T^{2} \)
43 \( 1 + (7.84e7 - 7.84e7i)T - 3.99e19iT^{2} \)
47 \( 1 + (4.58e9 + 4.58e9i)T + 1.16e20iT^{2} \)
53 \( 1 + (2.84e10 - 2.84e10i)T - 4.91e20iT^{2} \)
59 \( 1 - 6.01e10iT - 1.77e21T^{2} \)
61 \( 1 - 7.83e10T + 2.65e21T^{2} \)
67 \( 1 + (5.83e10 + 5.83e10i)T + 8.18e21iT^{2} \)
71 \( 1 - 2.48e11T + 1.64e22T^{2} \)
73 \( 1 + (-6.89e10 + 6.89e10i)T - 2.29e22iT^{2} \)
79 \( 1 - 4.47e11iT - 5.90e22T^{2} \)
83 \( 1 + (5.45e10 - 5.45e10i)T - 1.06e23iT^{2} \)
89 \( 1 - 1.68e11iT - 2.46e23T^{2} \)
97 \( 1 + (8.68e11 + 8.68e11i)T + 6.93e23iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.46253967930644062289544329148, −16.68565255113990437572373436068, −15.18650326668528516260126941751, −14.24753335938767928675919153385, −12.52931358222352873867357245394, −10.99767543537333077697600232193, −8.361753870197148771102325985956, −7.07026561414573856416064701330, −4.83927921602943592444033528064, −2.61641879015446702704345855663, 0.814999744957627247110107349177, 3.44048892740874915779642690196, 5.01729398150977995866673035171, 7.77465540652266369969953427612, 9.724078676782854618307583412182, 11.50779397084261905810088946036, 12.78647346869250342025986887871, 14.49106911253885997611115056813, 15.66305642248460394914176704440, 17.39949211907612192382884866349

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