Properties

Label 2-304-19.12-c2-0-4
Degree $2$
Conductor $304$
Sign $0.429 - 0.902i$
Analytic cond. $8.28340$
Root an. cond. $2.87808$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.275 + 0.158i)3-s + (0.5 + 0.866i)5-s + 2.89·7-s + (−4.44 + 7.70i)9-s + 5.10·11-s + (−0.151 − 0.0874i)13-s + (−0.275 − 0.158i)15-s + (5.94 + 10.3i)17-s + (3.34 + 18.7i)19-s + (−0.797 + 0.460i)21-s + (−8.52 + 14.7i)23-s + (12 − 20.7i)25-s − 5.68i·27-s + (38.5 + 22.2i)29-s + 31.1i·31-s + ⋯
L(s)  = 1  + (−0.0917 + 0.0529i)3-s + (0.100 + 0.173i)5-s + 0.414·7-s + (−0.494 + 0.856i)9-s + 0.463·11-s + (−0.0116 − 0.00672i)13-s + (−0.0183 − 0.0105i)15-s + (0.349 + 0.606i)17-s + (0.176 + 0.984i)19-s + (−0.0379 + 0.0219i)21-s + (−0.370 + 0.641i)23-s + (0.479 − 0.831i)25-s − 0.210i·27-s + (1.32 + 0.767i)29-s + 1.00i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 - 0.902i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.429 - 0.902i$
Analytic conductor: \(8.28340\)
Root analytic conductor: \(2.87808\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1),\ 0.429 - 0.902i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.30830 + 0.826148i\)
\(L(\frac12)\) \(\approx\) \(1.30830 + 0.826148i\)
\(L(2)\) 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 \( 1 + (-3.34 - 18.7i)T \)
good3 \( 1 + (0.275 - 0.158i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 2.89T + 49T^{2} \)
11 \( 1 - 5.10T + 121T^{2} \)
13 \( 1 + (0.151 + 0.0874i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-5.94 - 10.3i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (8.52 - 14.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-38.5 - 22.2i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 31.1iT - 961T^{2} \)
37 \( 1 - 21.9iT - 1.36e3T^{2} \)
41 \( 1 + (46.9 - 27.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-18.6 - 32.3i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-40.7 + 70.6i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-48.2 - 27.8i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-29.9 + 17.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-38.0 + 65.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (102. + 59.2i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (65.4 - 37.8i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (14.6 + 25.4i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-57.2 + 33.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 30.6T + 6.88e3T^{2} \)
89 \( 1 + (8.84 + 5.10i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (128. - 74.3i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−11.70717736158990026466755542517, −10.62778784941210281251925763041, −10.04742839108288629886557474045, −8.617190867174253233123163185763, −8.014969722305128419791670282199, −6.74901431973253637646504825098, −5.65995738231737544707615930747, −4.62459706923003704702182167972, −3.18783036029967824064938705486, −1.62418567435985871527786141012, 0.819888945744019031560038321134, 2.68350368831210221323304851382, 4.11590298658617140626371755730, 5.32611783762862693041791771433, 6.40208390160890462718742213172, 7.40648978018492637906392433871, 8.663642099316022112081176378606, 9.300775805039339410452455632263, 10.41714853220462668615289554696, 11.57809307881668243481781034532

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