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Properties

Label 2-304-19.7-c1-0-3
Degree $2$
Conductor $304$
Sign $0.910 - 0.412i$
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  + (0.5 + 0.866i)3-s + 4·7-s + (1 − 1.73i)9-s − 3·11-s + (−1 + 1.73i)13-s + (3 + 5.19i)17-s + (3.5 − 2.59i)19-s + (2 + 3.46i)21-s + (−3 + 5.19i)23-s + (2.5 − 4.33i)25-s + 5·27-s − 2·31-s + (−1.5 − 2.59i)33-s − 10·37-s − 1.99·39-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + 1.51·7-s + (0.333 − 0.577i)9-s − 0.904·11-s + (−0.277 + 0.480i)13-s + (0.727 + 1.26i)17-s + (0.802 − 0.596i)19-s + (0.436 + 0.755i)21-s + (−0.625 + 1.08i)23-s + (0.5 − 0.866i)25-s + 0.962·27-s − 0.359·31-s + (−0.261 − 0.452i)33-s − 1.64·37-s − 0.320·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.910 - 0.412i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.910 - 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56458 + 0.338148i\)
\(L(\frac12)\) \(\approx\) \(1.56458 + 0.338148i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-3.5 + 2.59i)T \)
good3 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.5 + 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.79046945920849194084307987192, −10.71847297110403078461802098912, −10.03159301912368317158478020777, −8.879975529068642090755983407217, −8.064793730236352487603491001056, −7.12523186046341795950573104520, −5.56460508320998101200922413237, −4.66394074691694157525008378116, −3.49310160595025062941920708005, −1.74890550306580313603717464941, 1.55470481856806376749536800444, 2.90543307958866180738477720057, 4.80639296637173689723380906051, 5.37100781141545302116005837049, 7.18390465856659408311806978343, 7.81492517044600848278144870102, 8.464498472816492501032971920398, 9.932640092240573728227139780063, 10.73359721735879179389445418935, 11.70267975460397978759753019787

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