Properties

Label 2-304-304.243-c1-0-27
Degree $2$
Conductor $304$
Sign $0.985 - 0.170i$
Analytic cond. $2.42745$
Root an. cond. $1.55802$
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  + (1.20 + 0.734i)2-s + (0.156 − 0.223i)3-s + (0.921 + 1.77i)4-s + (0.245 − 2.80i)5-s + (0.353 − 0.155i)6-s + (1.67 − 2.89i)7-s + (−0.189 + 2.82i)8-s + (1.00 + 2.74i)9-s + (2.35 − 3.20i)10-s + (−0.324 + 1.21i)11-s + (0.540 + 0.0717i)12-s + (−2.59 − 3.71i)13-s + (4.14 − 2.27i)14-s + (−0.587 − 0.493i)15-s + (−2.30 + 3.27i)16-s + (5.19 + 1.88i)17-s + ⋯
L(s)  = 1  + (0.854 + 0.519i)2-s + (0.0903 − 0.129i)3-s + (0.460 + 0.887i)4-s + (0.109 − 1.25i)5-s + (0.144 − 0.0633i)6-s + (0.631 − 1.09i)7-s + (−0.0669 + 0.997i)8-s + (0.333 + 0.916i)9-s + (0.744 − 1.01i)10-s + (−0.0977 + 0.364i)11-s + (0.156 + 0.0207i)12-s + (−0.720 − 1.02i)13-s + (1.10 − 0.607i)14-s + (−0.151 − 0.127i)15-s + (−0.575 + 0.817i)16-s + (1.25 + 0.458i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.170i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.985 - 0.170i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.985 - 0.170i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22380 + 0.191182i\)
\(L(\frac12)\) \(\approx\) \(2.22380 + 0.191182i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.20 - 0.734i)T \)
19 \( 1 + (3.37 - 2.76i)T \)
good3 \( 1 + (-0.156 + 0.223i)T + (-1.02 - 2.81i)T^{2} \)
5 \( 1 + (-0.245 + 2.80i)T + (-4.92 - 0.868i)T^{2} \)
7 \( 1 + (-1.67 + 2.89i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.324 - 1.21i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (2.59 + 3.71i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (-5.19 - 1.88i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (2.66 + 2.23i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (2.11 - 4.54i)T + (-18.6 - 22.2i)T^{2} \)
31 \( 1 + (3.99 - 6.91i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.00 - 4.00i)T + 37iT^{2} \)
41 \( 1 + (1.22 + 6.93i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (8.74 + 0.764i)T + (42.3 + 7.46i)T^{2} \)
47 \( 1 + (1.80 + 4.96i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (1.08 + 12.3i)T + (-52.1 + 9.20i)T^{2} \)
59 \( 1 + (2.07 - 0.965i)T + (37.9 - 45.1i)T^{2} \)
61 \( 1 + (0.184 + 2.10i)T + (-60.0 + 10.5i)T^{2} \)
67 \( 1 + (-0.109 - 0.0512i)T + (43.0 + 51.3i)T^{2} \)
71 \( 1 + (2.06 + 2.45i)T + (-12.3 + 69.9i)T^{2} \)
73 \( 1 + (-16.3 + 2.87i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (-2.21 - 12.5i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-1.65 - 6.17i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-0.512 + 2.90i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (2.03 - 5.58i)T + (-74.3 - 62.3i)T^{2} \)
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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

−12.31723030573826351379261887096, −10.80606802130683801864030130159, −10.07672115907200072856172487519, −8.311261173184684559223585280443, −7.965591254569976363538044184027, −6.97661848539560214091682670025, −5.30820819648095118289145979244, −4.90834502080539109520502998471, −3.71841482595206330987853013626, −1.73915880319466756139854964970, 2.12381054915162968876890970987, 3.13547144296029432419292212544, 4.39321339436390081209557280247, 5.73173270497926273850409695210, 6.51014949700399261826756759036, 7.61318450944202446686774682582, 9.294019373533615540806334841040, 9.873277502797325532716964215735, 11.11680929506194782421393459041, 11.66973343562248383147957792724

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