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Properties

Label 2-308-11.4-c1-0-5
Degree $2$
Conductor $308$
Sign $-0.642 + 0.766i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 3.07i)3-s + (−1.61 + 1.17i)5-s + (−0.309 − 0.951i)7-s + (−6.04 − 4.39i)9-s + (2.19 − 2.48i)11-s + (−2.61 − 1.90i)13-s + (2 + 6.15i)15-s + (3.23 − 2.35i)17-s + (−0.618 + 1.90i)19-s − 3.23·21-s + 0.618·23-s + (−0.309 + 0.951i)25-s + (−11.7 + 8.50i)27-s + (−0.263 − 0.812i)29-s + (3.61 + 2.62i)31-s + ⋯
L(s)  = 1  + (0.577 − 1.77i)3-s + (−0.723 + 0.525i)5-s + (−0.116 − 0.359i)7-s + (−2.01 − 1.46i)9-s + (0.660 − 0.750i)11-s + (−0.726 − 0.527i)13-s + (0.516 + 1.58i)15-s + (0.784 − 0.570i)17-s + (−0.141 + 0.436i)19-s − 0.706·21-s + 0.128·23-s + (−0.0618 + 0.190i)25-s + (−2.25 + 1.63i)27-s + (−0.0490 − 0.150i)29-s + (0.649 + 0.472i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $-0.642 + 0.766i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ -0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.526406 - 1.12847i\)
\(L(\frac12)\) \(\approx\) \(0.526406 - 1.12847i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-2.19 + 2.48i)T \)
good3 \( 1 + (-1 + 3.07i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (1.61 - 1.17i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (2.61 + 1.90i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.23 + 2.35i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.618 - 1.90i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 0.618T + 23T^{2} \)
29 \( 1 + (0.263 + 0.812i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.61 - 2.62i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.04 - 9.37i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.61 + 11.1i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 + (-3.09 + 9.51i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.78 - 5.65i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.14 + 3.52i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.381 - 0.277i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 0.381T + 67T^{2} \)
71 \( 1 + (1.54 - 1.12i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.52 + 13.9i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.92 - 3.57i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (12.0 - 8.78i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 6.76T + 89T^{2} \)
97 \( 1 + (-12.4 - 9.06i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.84127166563422548373774879425, −10.60798161907394083290085194552, −9.197861781217209399705654525591, −8.147584271986705331840373213454, −7.48235719840982172211784867013, −6.82010672324734605582715153679, −5.72970822304745179465034495623, −3.63870223819977626283936472493, −2.66396132816344527912412319426, −0.892262277878367080252750852680, 2.67938631635799050530584678710, 4.14025162933058475337135901628, 4.48246664919288151282031000848, 5.78750112228799382554800837272, 7.54010240415491691440529433101, 8.539841452101099138972822450480, 9.362921929705556450966508471645, 9.886776502821215447546746280384, 11.01539094545031490630140272195, 11.87187220332522575998880025122

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