Properties

Label 2-10-5.3-c14-0-5
Degree $2$
Conductor $10$
Sign $-0.920 - 0.390i$
Analytic cond. $12.4328$
Root an. cond. $3.52603$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−64 + 64i)2-s + (−1.76e3 − 1.76e3i)3-s − 8.19e3i·4-s + (4.62e4 − 6.29e4i)5-s + 2.25e5·6-s + (−5.96e4 + 5.96e4i)7-s + (5.24e5 + 5.24e5i)8-s + 1.41e6i·9-s + (1.07e6 + 6.98e6i)10-s − 4.64e6·11-s + (−1.44e7 + 1.44e7i)12-s + (−4.85e7 − 4.85e7i)13-s − 7.63e6i·14-s + (−1.92e8 + 2.95e7i)15-s − 6.71e7·16-s + (−4.70e8 + 4.70e8i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.805 − 0.805i)3-s − 0.5i·4-s + (0.591 − 0.806i)5-s + 0.805·6-s + (−0.0724 + 0.0724i)7-s + (0.250 + 0.250i)8-s + 0.296i·9-s + (0.107 + 0.698i)10-s − 0.238·11-s + (−0.402 + 0.402i)12-s + (−0.773 − 0.773i)13-s − 0.0724i·14-s + (−1.12 + 0.172i)15-s − 0.250·16-s + (−1.14 + 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.920 - 0.390i$
Analytic conductor: \(12.4328\)
Root analytic conductor: \(3.52603\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :7),\ -0.920 - 0.390i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.0253527 + 0.124646i\)
\(L(\frac12)\) \(\approx\) \(0.0253527 + 0.124646i\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (64 - 64i)T \)
5 \( 1 + (-4.62e4 + 6.29e4i)T \)
good3 \( 1 + (1.76e3 + 1.76e3i)T + 4.78e6iT^{2} \)
7 \( 1 + (5.96e4 - 5.96e4i)T - 6.78e11iT^{2} \)
11 \( 1 + 4.64e6T + 3.79e14T^{2} \)
13 \( 1 + (4.85e7 + 4.85e7i)T + 3.93e15iT^{2} \)
17 \( 1 + (4.70e8 - 4.70e8i)T - 1.68e17iT^{2} \)
19 \( 1 - 7.64e8iT - 7.99e17T^{2} \)
23 \( 1 + (-1.43e9 - 1.43e9i)T + 1.15e19iT^{2} \)
29 \( 1 - 3.55e9iT - 2.97e20T^{2} \)
31 \( 1 - 4.07e10T + 7.56e20T^{2} \)
37 \( 1 + (1.25e11 - 1.25e11i)T - 9.01e21iT^{2} \)
41 \( 1 + 1.93e11T + 3.79e22T^{2} \)
43 \( 1 + (2.99e11 + 2.99e11i)T + 7.38e22iT^{2} \)
47 \( 1 + (-4.49e11 + 4.49e11i)T - 2.56e23iT^{2} \)
53 \( 1 + (8.73e11 + 8.73e11i)T + 1.37e24iT^{2} \)
59 \( 1 + 2.30e12iT - 6.19e24T^{2} \)
61 \( 1 - 1.81e12T + 9.87e24T^{2} \)
67 \( 1 + (5.96e12 - 5.96e12i)T - 3.67e25iT^{2} \)
71 \( 1 - 5.96e12T + 8.27e25T^{2} \)
73 \( 1 + (7.19e11 + 7.19e11i)T + 1.22e26iT^{2} \)
79 \( 1 + 3.29e12iT - 3.68e26T^{2} \)
83 \( 1 + (6.80e12 + 6.80e12i)T + 7.36e26iT^{2} \)
89 \( 1 + 3.40e13iT - 1.95e27T^{2} \)
97 \( 1 + (7.87e13 - 7.87e13i)T - 6.52e27iT^{2} \)
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

−17.11140427063895259462304083769, −15.36195143466284017641259753739, −13.34276562414128841903193687214, −12.15982491619917716643005231494, −10.18654748253584399714000290982, −8.394487767579140120516758800473, −6.59106126734779167303545087104, −5.29408584535899818843583703261, −1.56967147870654896428590433510, −0.07033703289042689368084490191, 2.49849083420840402233099454941, 4.77120784739518636035285519998, 6.85402815199494331227929449361, 9.397543256642367626990247575327, 10.56461978604801446536959793434, 11.59717014289871328957556814619, 13.69636619831841176957843840594, 15.57653927272628597691951429369, 16.96506739253941094161451927645, 17.96570195901098597583292085136

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