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Properties

Label 2-308-77.37-c1-0-6
Degree $2$
Conductor $308$
Sign $-0.577 + 0.816i$
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  + (−1.23 + 0.262i)3-s + (−0.367 − 3.50i)5-s + (0.745 + 2.53i)7-s + (−1.28 + 0.573i)9-s + (0.496 − 3.27i)11-s + (−5.68 − 4.13i)13-s + (1.37 + 4.22i)15-s + (0.124 + 0.0554i)17-s + (−3.02 − 3.36i)19-s + (−1.58 − 2.93i)21-s + (0.770 − 1.33i)23-s + (−7.22 + 1.53i)25-s + (4.49 − 3.26i)27-s + (−0.662 − 2.04i)29-s + (−0.0801 + 0.762i)31-s + ⋯
L(s)  = 1  + (−0.712 + 0.151i)3-s + (−0.164 − 1.56i)5-s + (0.281 + 0.959i)7-s + (−0.429 + 0.191i)9-s + (0.149 − 0.988i)11-s + (−1.57 − 1.14i)13-s + (0.354 + 1.08i)15-s + (0.0301 + 0.0134i)17-s + (−0.694 − 0.770i)19-s + (−0.345 − 0.640i)21-s + (0.160 − 0.278i)23-s + (−1.44 + 0.307i)25-s + (0.865 − 0.629i)27-s + (−0.123 − 0.378i)29-s + (−0.0144 + 0.137i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ -0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.282080 - 0.544906i\)
\(L(\frac12)\) \(\approx\) \(0.282080 - 0.544906i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.745 - 2.53i)T \)
11 \( 1 + (-0.496 + 3.27i)T \)
good3 \( 1 + (1.23 - 0.262i)T + (2.74 - 1.22i)T^{2} \)
5 \( 1 + (0.367 + 3.50i)T + (-4.89 + 1.03i)T^{2} \)
13 \( 1 + (5.68 + 4.13i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.124 - 0.0554i)T + (11.3 + 12.6i)T^{2} \)
19 \( 1 + (3.02 + 3.36i)T + (-1.98 + 18.8i)T^{2} \)
23 \( 1 + (-0.770 + 1.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.662 + 2.04i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.0801 - 0.762i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (-2.61 - 0.556i)T + (33.8 + 15.0i)T^{2} \)
41 \( 1 + (-2.06 + 6.36i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.40T + 43T^{2} \)
47 \( 1 + (-4.98 - 5.54i)T + (-4.91 + 46.7i)T^{2} \)
53 \( 1 + (1.12 - 10.7i)T + (-51.8 - 11.0i)T^{2} \)
59 \( 1 + (2.24 - 2.48i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (0.0132 + 0.125i)T + (-59.6 + 12.6i)T^{2} \)
67 \( 1 + (-4.39 - 7.61i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.94 + 3.59i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-10.0 + 11.1i)T + (-7.63 - 72.6i)T^{2} \)
79 \( 1 + (-6.92 + 3.08i)T + (52.8 - 58.7i)T^{2} \)
83 \( 1 + (6.56 - 4.76i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-7.57 + 13.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.61 + 6.98i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.53428991519048173243936959866, −10.60105977133254498416221605150, −9.287415881306250287689659226821, −8.607371091957525928803352792197, −7.79637844058091195613813914619, −6.00710949999057903547337632923, −5.28914568089900236732856605546, −4.63139125140298897768925973037, −2.62598603122645913666021579858, −0.46519864889948653582168416803, 2.22964498469768770228607632669, 3.78099600337908795959031341655, 4.96654125518292848088305143610, 6.54730575533727760860234506770, 6.94201987893355145373352430998, 7.83167958080478694702141298256, 9.576400308188744120173814826550, 10.29457222665088312153813772559, 11.16964532293724442814900433524, 11.77208959054421279812129686130

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