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Properties

Label 2-308-11.2-c2-0-7
Degree $2$
Conductor $308$
Sign $0.999 + 0.0387i$
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  + (3.43 − 2.49i)3-s + (1.71 + 5.29i)5-s + (−1.55 + 2.14i)7-s + (2.78 − 8.57i)9-s + (10.9 + 0.184i)11-s + (4.24 + 1.37i)13-s + (19.1 + 13.8i)15-s + (17.6 − 5.72i)17-s + (8.75 + 12.0i)19-s + 11.2i·21-s − 39.7·23-s + (−4.81 + 3.50i)25-s + (−0.0188 − 0.0580i)27-s + (20.9 − 28.7i)29-s + (0.350 − 1.07i)31-s + ⋯
L(s)  = 1  + (1.14 − 0.831i)3-s + (0.343 + 1.05i)5-s + (−0.222 + 0.305i)7-s + (0.309 − 0.952i)9-s + (0.999 + 0.0167i)11-s + (0.326 + 0.106i)13-s + (1.27 + 0.925i)15-s + (1.03 − 0.336i)17-s + (0.460 + 0.633i)19-s + 0.534i·21-s − 1.72·23-s + (−0.192 + 0.140i)25-s + (−0.000698 − 0.00214i)27-s + (0.720 − 0.991i)29-s + (0.0113 − 0.0347i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0387i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.999 + 0.0387i$
Analytic conductor: \(8.39239\)
Root analytic conductor: \(2.89696\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1),\ 0.999 + 0.0387i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.59102 - 0.0502475i\)
\(L(\frac12)\) \(\approx\) \(2.59102 - 0.0502475i\)
\(L(2)\) 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 \( 1 + (-10.9 - 0.184i)T \)
good3 \( 1 + (-3.43 + 2.49i)T + (2.78 - 8.55i)T^{2} \)
5 \( 1 + (-1.71 - 5.29i)T + (-20.2 + 14.6i)T^{2} \)
13 \( 1 + (-4.24 - 1.37i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (-17.6 + 5.72i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-8.75 - 12.0i)T + (-111. + 343. i)T^{2} \)
23 \( 1 + 39.7T + 529T^{2} \)
29 \( 1 + (-20.9 + 28.7i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-0.350 + 1.07i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (3.05 + 2.22i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (40.0 + 55.1i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 25.6iT - 1.84e3T^{2} \)
47 \( 1 + (64.9 - 47.1i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (10.2 - 31.5i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (-63.8 - 46.4i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (27.2 - 8.84i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 - 10.3T + 4.48e3T^{2} \)
71 \( 1 + (12.1 + 37.3i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-51.0 + 70.2i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (100. + 32.5i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (69.8 - 22.7i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 + 80.5T + 7.92e3T^{2} \)
97 \( 1 + (-13.4 + 41.2i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−11.73797342883056884104528443653, −10.30766278294131859658711451081, −9.578538225847582674426297298334, −8.491690089824206653294478253430, −7.64899622218922171938356519127, −6.71524650064276352316569830559, −5.88086590873650318335383139623, −3.78306374907928524559902863446, −2.82372364987014650764814470149, −1.66969917174148176013589172970, 1.40807804287363240209211090566, 3.22178503624712884163019778044, 4.12598487953387993972242618094, 5.21950331433887885356936967212, 6.58999910596732794293556282835, 8.147658155518008602990607567875, 8.650317011858437504535774296443, 9.706839899458660021487057893842, 9.987726621624402613360175631141, 11.51748272194453890319712467024

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