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Properties

Label 2-304-304.27-c1-0-33
Degree $2$
Conductor $304$
Sign $-0.792 + 0.610i$
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  + (0.920 − 1.07i)2-s + (−0.668 − 2.49i)3-s + (−0.304 − 1.97i)4-s + (1.13 − 0.304i)5-s + (−3.29 − 1.57i)6-s + 4.39·7-s + (−2.40 − 1.49i)8-s + (−3.17 + 1.83i)9-s + (0.719 − 1.49i)10-s + (0.949 − 0.949i)11-s + (−4.72 + 2.08i)12-s + (−1.39 + 5.21i)13-s + (4.04 − 4.71i)14-s + (−1.51 − 2.62i)15-s + (−3.81 + 1.20i)16-s + (−0.965 + 1.67i)17-s + ⋯
L(s)  = 1  + (0.651 − 0.758i)2-s + (−0.385 − 1.44i)3-s + (−0.152 − 0.988i)4-s + (0.507 − 0.136i)5-s + (−1.34 − 0.644i)6-s + 1.66·7-s + (−0.849 − 0.528i)8-s + (−1.05 + 0.611i)9-s + (0.227 − 0.474i)10-s + (0.286 − 0.286i)11-s + (−1.36 + 0.600i)12-s + (−0.387 + 1.44i)13-s + (1.08 − 1.26i)14-s + (−0.391 − 0.678i)15-s + (−0.953 + 0.300i)16-s + (−0.234 + 0.405i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.792 + 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.792 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.792 + 0.610i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ -0.792 + 0.610i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.600132 - 1.76301i\)
\(L(\frac12)\) \(\approx\) \(0.600132 - 1.76301i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.920 + 1.07i)T \)
19 \( 1 + (1.70 - 4.01i)T \)
good3 \( 1 + (0.668 + 2.49i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (-1.13 + 0.304i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 - 4.39T + 7T^{2} \)
11 \( 1 + (-0.949 + 0.949i)T - 11iT^{2} \)
13 \( 1 + (1.39 - 5.21i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.965 - 1.67i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.82 - 6.62i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.06 + 7.69i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + 0.0210T + 31T^{2} \)
37 \( 1 + (0.0844 - 0.0844i)T - 37iT^{2} \)
41 \( 1 + (2.05 - 3.55i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.99 + 7.45i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-6.68 + 3.85i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.64 + 1.77i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-10.3 + 2.78i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (10.4 + 2.80i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.09 + 0.561i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (8.87 + 5.12i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.0766 + 0.0442i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.95 - 8.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.53 + 1.53i)T + 83iT^{2} \)
89 \( 1 + (0.966 + 1.67i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.24 - 1.87i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.78047515646009910661630350818, −10.92266212655176934367680528168, −9.588385520772271324116452739822, −8.433642110548498043128117446200, −7.31499624541237581057726807095, −6.22659008650954570296980344443, −5.36506250205882113304921316838, −4.17269321500730459804155847041, −1.99218182673004247651921285551, −1.52413230755739186836530044937, 2.82279614381486808633130216829, 4.48275247469422587539261049609, 4.88185687588300965490137505659, 5.75965064656284711859040555698, 7.15701865609521578521988167786, 8.330551990072747271149243615711, 9.143447955238309612334735901471, 10.44792391914811694517953258378, 11.01039612414910219304211713554, 12.02883021693897916282123692236

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