↑

Properties

Label 2-304-16.13-c1-0-19
Degree $2$
Conductor $304$
Sign $0.999 - 0.0328i$
Analytic cond. $2.42745$
Root an. cond. $1.55802$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.399 + 1.35i)2-s + (0.464 + 0.464i)3-s + (−1.68 − 1.08i)4-s + (0.625 − 0.625i)5-s + (−0.815 + 0.444i)6-s − 3.89i·7-s + (2.14 − 1.84i)8-s − 2.56i·9-s + (0.599 + 1.09i)10-s + (2.34 − 2.34i)11-s + (−0.277 − 1.28i)12-s + (2.42 + 2.42i)13-s + (5.28 + 1.55i)14-s + 0.580·15-s + (1.65 + 3.64i)16-s − 7.26·17-s + ⋯
L(s)  = 1  + (−0.282 + 0.959i)2-s + (0.267 + 0.267i)3-s + (−0.840 − 0.541i)4-s + (0.279 − 0.279i)5-s + (−0.332 + 0.181i)6-s − 1.47i·7-s + (0.757 − 0.653i)8-s − 0.856i·9-s + (0.189 + 0.347i)10-s + (0.706 − 0.706i)11-s + (−0.0800 − 0.370i)12-s + (0.673 + 0.673i)13-s + (1.41 + 0.416i)14-s + 0.150·15-s + (0.412 + 0.910i)16-s − 1.76·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.999 - 0.0328i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.999 - 0.0328i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17309 + 0.0192503i\)
\(L(\frac12)\) \(\approx\) \(1.17309 + 0.0192503i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.399 - 1.35i)T \)
19 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-0.464 - 0.464i)T + 3iT^{2} \)
5 \( 1 + (-0.625 + 0.625i)T - 5iT^{2} \)
7 \( 1 + 3.89iT - 7T^{2} \)
11 \( 1 + (-2.34 + 2.34i)T - 11iT^{2} \)
13 \( 1 + (-2.42 - 2.42i)T + 13iT^{2} \)
17 \( 1 + 7.26T + 17T^{2} \)
23 \( 1 + 0.368iT - 23T^{2} \)
29 \( 1 + (3.32 + 3.32i)T + 29iT^{2} \)
31 \( 1 - 5.74T + 31T^{2} \)
37 \( 1 + (-5.87 + 5.87i)T - 37iT^{2} \)
41 \( 1 - 10.5iT - 41T^{2} \)
43 \( 1 + (3.58 - 3.58i)T - 43iT^{2} \)
47 \( 1 - 11.4T + 47T^{2} \)
53 \( 1 + (5.37 - 5.37i)T - 53iT^{2} \)
59 \( 1 + (-4.48 + 4.48i)T - 59iT^{2} \)
61 \( 1 + (-6.06 - 6.06i)T + 61iT^{2} \)
67 \( 1 + (7.00 + 7.00i)T + 67iT^{2} \)
71 \( 1 + 2.78iT - 71T^{2} \)
73 \( 1 + 9.31iT - 73T^{2} \)
79 \( 1 - 2.88T + 79T^{2} \)
83 \( 1 + (3.05 + 3.05i)T + 83iT^{2} \)
89 \( 1 - 14.7iT - 89T^{2} \)
97 \( 1 + 0.337T + 97T^{2} \)
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

−11.48821680657620266939391812605, −10.63003752118254020035433452471, −9.413877673550680388200808833228, −9.042973009446230396770943725867, −7.891223148431087504134585770138, −6.70728853950682424363809437204, −6.18117654169850183128517761675, −4.44625712733104900292542656874, −3.83117012700749439009170060703, −1.02579298064225913968474539553, 1.96820144648538897299219817221, 2.72876041385445619691655079104, 4.38676926998325886753963047382, 5.59607591977636995405394927689, 6.96463745927335576633857542923, 8.400781595848794094165531109412, 8.814687876333458525310159439926, 9.882745820014295444746512152529, 10.86594722633432434702957206570, 11.66540111721419410800111592478

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