↑

Properties

Label 2-308-11.5-c1-0-5
Degree $2$
Conductor $308$
Sign $-0.929 + 0.369i$
Analytic cond. $2.45939$
Root an. cond. $1.56824$
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  + (−2.11 − 1.53i)3-s + (1.17 − 3.61i)5-s + (0.809 − 0.587i)7-s + (1.19 + 3.66i)9-s + (−2.45 + 2.23i)11-s + (−0.482 − 1.48i)13-s + (−8.04 + 5.84i)15-s + (−0.107 + 0.329i)17-s + (−5.77 − 4.19i)19-s − 2.61·21-s + 4.61·23-s + (−7.61 − 5.53i)25-s + (0.690 − 2.12i)27-s + (−2.04 + 1.48i)29-s + (0.583 + 1.79i)31-s + ⋯
L(s)  = 1  + (−1.22 − 0.888i)3-s + (0.524 − 1.61i)5-s + (0.305 − 0.222i)7-s + (0.396 + 1.22i)9-s + (−0.739 + 0.672i)11-s + (−0.133 − 0.411i)13-s + (−2.07 + 1.50i)15-s + (−0.0259 + 0.0799i)17-s + (−1.32 − 0.963i)19-s − 0.571·21-s + 0.962·23-s + (−1.52 − 1.10i)25-s + (0.132 − 0.409i)27-s + (−0.379 + 0.275i)29-s + (0.104 + 0.322i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $-0.929 + 0.369i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ -0.929 + 0.369i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.139720 - 0.730143i\)
\(L(\frac12)\) \(\approx\) \(0.139720 - 0.730143i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (2.45 - 2.23i)T \)
good3 \( 1 + (2.11 + 1.53i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-1.17 + 3.61i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (0.482 + 1.48i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.107 - 0.329i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.77 + 4.19i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 4.61T + 23T^{2} \)
29 \( 1 + (2.04 - 1.48i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.583 - 1.79i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-8.23 + 5.98i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-2.11 - 1.53i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 12.9T + 43T^{2} \)
47 \( 1 + (-5.25 - 3.81i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.955 - 2.94i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.04 + 3.66i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.42 + 10.5i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 9.86T + 67T^{2} \)
71 \( 1 + (-3.09 + 9.53i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.444 - 0.323i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.69 + 11.3i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.65 + 14.3i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 + (-1.08 - 3.33i)T + (-78.4 + 57.0i)T^{2} \)
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

−11.40160486330733073989237890680, −10.57192719600926754529842836042, −9.379542994014975966389867441371, −8.355168380571113338908422456485, −7.33030574823700847477632882969, −6.23043299604839677907952297862, −5.17990996691289874034191270022, −4.68526127874438976277231465138, −1.97433622026229544446406717747, −0.61928654239752584902197355203, 2.53265258653519666574599587352, 3.96513710255660529511597435667, 5.32467275279899865054218574535, 6.09633806056709031567706336916, 6.91426106697302404872235603221, 8.338499992550775495580229065457, 9.862368470406472653420040453547, 10.32876904920413207139492875541, 11.18091196513700058652258094779, 11.51669604765695681387432179854

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