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Properties

Label 2-308-11.6-c2-0-6
Degree $2$
Conductor $308$
Sign $-0.329 + 0.944i$
Analytic cond. $8.39239$
Root an. cond. $2.89696$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.23 − 2.34i)3-s + (−2.54 + 7.82i)5-s + (1.55 + 2.14i)7-s + (2.15 + 6.63i)9-s + (3.85 − 10.3i)11-s + (4.32 − 1.40i)13-s + (26.6 − 19.3i)15-s + (−23.7 − 7.71i)17-s + (6.86 − 9.44i)19-s − 10.5i·21-s + 9.61·23-s + (−34.5 − 25.0i)25-s + (−2.49 + 7.68i)27-s + (−22.3 − 30.7i)29-s + (−7.53 − 23.1i)31-s + ⋯
L(s)  = 1  + (−1.07 − 0.783i)3-s + (−0.508 + 1.56i)5-s + (0.222 + 0.305i)7-s + (0.239 + 0.737i)9-s + (0.350 − 0.936i)11-s + (0.332 − 0.107i)13-s + (1.77 − 1.28i)15-s + (−1.39 − 0.453i)17-s + (0.361 − 0.497i)19-s − 0.503i·21-s + 0.418·23-s + (−1.38 − 1.00i)25-s + (−0.0924 + 0.284i)27-s + (−0.769 − 1.05i)29-s + (−0.243 − 0.748i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $-0.329 + 0.944i$
Analytic conductor: \(8.39239\)
Root analytic conductor: \(2.89696\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1),\ -0.329 + 0.944i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.337206 - 0.474757i\)
\(L(\frac12)\) \(\approx\) \(0.337206 - 0.474757i\)
\(L(2)\) 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 \( 1 + (-1.55 - 2.14i)T \)
11 \( 1 + (-3.85 + 10.3i)T \)
good3 \( 1 + (3.23 + 2.34i)T + (2.78 + 8.55i)T^{2} \)
5 \( 1 + (2.54 - 7.82i)T + (-20.2 - 14.6i)T^{2} \)
13 \( 1 + (-4.32 + 1.40i)T + (136. - 99.3i)T^{2} \)
17 \( 1 + (23.7 + 7.71i)T + (233. + 169. i)T^{2} \)
19 \( 1 + (-6.86 + 9.44i)T + (-111. - 343. i)T^{2} \)
23 \( 1 - 9.61T + 529T^{2} \)
29 \( 1 + (22.3 + 30.7i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (7.53 + 23.1i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (-36.8 + 26.7i)T + (423. - 1.30e3i)T^{2} \)
41 \( 1 + (-5.08 + 6.99i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 + 23.8iT - 1.84e3T^{2} \)
47 \( 1 + (45.5 + 33.1i)T + (682. + 2.10e3i)T^{2} \)
53 \( 1 + (6.50 + 20.0i)T + (-2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (-46.1 + 33.5i)T + (1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (-44.7 - 14.5i)T + (3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 - 70.4T + 4.48e3T^{2} \)
71 \( 1 + (-35.2 + 108. i)T + (-4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (-17.7 - 24.3i)T + (-1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (107. - 34.8i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (-70.4 - 22.8i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 + 140.T + 7.92e3T^{2} \)
97 \( 1 + (-39.8 - 122. i)T + (-7.61e3 + 5.53e3i)T^{2} \)
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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

−11.36061166967980027417886908978, −10.84046088020109546764577061481, −9.369263825525593271556014415152, −8.039941181962952586640795476184, −6.98617840948018637283786572613, −6.46520434635826794735663908325, −5.51030714736311773314371007074, −3.80952501815611706930504116422, −2.42071828489384388810238238967, −0.34792286927126442746475702961, 1.35043555361785785574820177540, 4.06622744939206235539226049754, 4.64716933711665638336727057331, 5.45630342811819523727151233338, 6.77253272331720061148425502875, 8.108119267387091251975102503643, 9.050092120499190958398785147531, 9.897211639656402367200764635626, 11.04438050448714439930947713403, 11.59632205046330477328855139690

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