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Properties

Label 2-304-19.18-c6-0-16
Degree $2$
Conductor $304$
Sign $-0.945 - 0.326i$
Analytic cond. $69.9364$
Root an. cond. $8.36280$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 33.0i·3-s + 159.·5-s − 445.·7-s − 365.·9-s + 1.70e3·11-s + 439. i·13-s + 5.28e3i·15-s + 569.·17-s + (−6.48e3 − 2.24e3i)19-s − 1.47e4i·21-s + 2.74e3·23-s + 9.85e3·25-s + 1.20e4i·27-s + 2.69e4i·29-s + 2.98e4i·31-s + ⋯
L(s)  = 1  + 1.22i·3-s + 1.27·5-s − 1.29·7-s − 0.501·9-s + 1.27·11-s + 0.199i·13-s + 1.56i·15-s + 0.115·17-s + (−0.945 − 0.326i)19-s − 1.59i·21-s + 0.225·23-s + 0.630·25-s + 0.611i·27-s + 1.10i·29-s + 1.00i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.945 - 0.326i$
Analytic conductor: \(69.9364\)
Root analytic conductor: \(8.36280\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :3),\ -0.945 - 0.326i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.834962601\)
\(L(\frac12)\) \(\approx\) \(1.834962601\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (6.48e3 + 2.24e3i)T \)
good3 \( 1 - 33.0iT - 729T^{2} \)
5 \( 1 - 159.T + 1.56e4T^{2} \)
7 \( 1 + 445.T + 1.17e5T^{2} \)
11 \( 1 - 1.70e3T + 1.77e6T^{2} \)
13 \( 1 - 439. iT - 4.82e6T^{2} \)
17 \( 1 - 569.T + 2.41e7T^{2} \)
23 \( 1 - 2.74e3T + 1.48e8T^{2} \)
29 \( 1 - 2.69e4iT - 5.94e8T^{2} \)
31 \( 1 - 2.98e4iT - 8.87e8T^{2} \)
37 \( 1 - 5.25e4iT - 2.56e9T^{2} \)
41 \( 1 + 2.73e4iT - 4.75e9T^{2} \)
43 \( 1 + 6.74e4T + 6.32e9T^{2} \)
47 \( 1 - 6.63e4T + 1.07e10T^{2} \)
53 \( 1 - 5.99e4iT - 2.21e10T^{2} \)
59 \( 1 - 3.96e5iT - 4.21e10T^{2} \)
61 \( 1 + 7.35e4T + 5.15e10T^{2} \)
67 \( 1 + 1.60e5iT - 9.04e10T^{2} \)
71 \( 1 + 2.59e5iT - 1.28e11T^{2} \)
73 \( 1 - 2.91e5T + 1.51e11T^{2} \)
79 \( 1 + 9.17e5iT - 2.43e11T^{2} \)
83 \( 1 + 6.81e5T + 3.26e11T^{2} \)
89 \( 1 - 2.42e5iT - 4.96e11T^{2} \)
97 \( 1 - 1.24e6iT - 8.32e11T^{2} \)
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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

−10.62619372890889109998693134990, −10.15066384511769323312179315441, −9.195603885449953309826501325570, −8.996952212172617876947342054237, −6.83442014699664157451725919640, −6.23722886647222496868726110175, −5.08500608803023728076132143688, −3.95027909570107355898042371286, −2.96713815746860534604477510237, −1.45142551055740988145347669295, 0.41991626467644076782994981646, 1.60533898696802815167841681975, 2.49765676486044912076294635390, 3.94921432942633070227899912821, 5.83033622281488699249438577011, 6.36171215670883187743880279880, 7.01906490026660394681993150297, 8.347563794420192708533120710224, 9.528970341658553681098352257652, 9.930316765098818602083841483306

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