Properties

Label 2-304-304.245-c1-0-9
Degree $2$
Conductor $304$
Sign $-0.642 - 0.766i$
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  + (−0.582 + 1.28i)2-s + (0.303 − 0.0265i)3-s + (−1.32 − 1.50i)4-s + (2.27 + 1.05i)5-s + (−0.142 + 0.405i)6-s + (−3.51 + 2.03i)7-s + (2.70 − 0.826i)8-s + (−2.86 + 0.504i)9-s + (−2.69 + 2.31i)10-s + (4.66 − 1.25i)11-s + (−0.440 − 0.420i)12-s + (−0.516 + 5.90i)13-s + (−0.566 − 5.71i)14-s + (0.717 + 0.260i)15-s + (−0.511 + 3.96i)16-s + (−1.21 + 6.89i)17-s + ⋯
L(s)  = 1  + (−0.412 + 0.911i)2-s + (0.174 − 0.0153i)3-s + (−0.660 − 0.750i)4-s + (1.01 + 0.474i)5-s + (−0.0581 + 0.165i)6-s + (−1.32 + 0.767i)7-s + (0.956 − 0.292i)8-s + (−0.954 + 0.168i)9-s + (−0.850 + 0.730i)10-s + (1.40 − 0.377i)11-s + (−0.127 − 0.121i)12-s + (−0.143 + 1.63i)13-s + (−0.151 − 1.52i)14-s + (0.185 + 0.0673i)15-s + (−0.127 + 0.991i)16-s + (−0.294 + 1.67i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ -0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.419499 + 0.898445i\)
\(L(\frac12)\) \(\approx\) \(0.419499 + 0.898445i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.582 - 1.28i)T \)
19 \( 1 + (-2.13 + 3.80i)T \)
good3 \( 1 + (-0.303 + 0.0265i)T + (2.95 - 0.520i)T^{2} \)
5 \( 1 + (-2.27 - 1.05i)T + (3.21 + 3.83i)T^{2} \)
7 \( 1 + (3.51 - 2.03i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.66 + 1.25i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.516 - 5.90i)T + (-12.8 - 2.25i)T^{2} \)
17 \( 1 + (1.21 - 6.89i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (0.212 - 0.583i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.999 - 0.699i)T + (9.91 + 27.2i)T^{2} \)
31 \( 1 + (-0.300 - 0.519i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.965 + 0.965i)T + 37iT^{2} \)
41 \( 1 + (-3.63 + 4.33i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (1.14 + 0.534i)T + (27.6 + 32.9i)T^{2} \)
47 \( 1 + (1.42 + 8.06i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-2.84 - 6.10i)T + (-34.0 + 40.6i)T^{2} \)
59 \( 1 + (0.939 - 0.657i)T + (20.1 - 55.4i)T^{2} \)
61 \( 1 + (-12.4 + 5.82i)T + (39.2 - 46.7i)T^{2} \)
67 \( 1 + (-0.923 - 0.646i)T + (22.9 + 62.9i)T^{2} \)
71 \( 1 + (-1.24 - 3.41i)T + (-54.3 + 45.6i)T^{2} \)
73 \( 1 + (-7.52 + 8.97i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (-10.4 - 8.78i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-1.01 - 0.273i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (0.154 + 0.184i)T + (-15.4 + 87.6i)T^{2} \)
97 \( 1 + (0.344 - 1.95i)T + (-91.1 - 33.1i)T^{2} \)
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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

−12.06011176219837070442715232517, −10.92801388268242817198840337505, −9.713482494724583936946028520936, −9.160514073252215107226807530208, −8.596493973551275024941431950866, −6.72632693426738814215209143192, −6.42144669261561392513490781890, −5.58941605526123840380687420033, −3.80475534337691606626450030072, −2.11211040708281917347882368332, 0.840719206285703441165359522080, 2.72889129275841608650099767152, 3.67746609347703005777343672438, 5.25163346815230096192096980088, 6.46716535224690182055136791507, 7.73570723116747444703002359725, 9.014732865481345660168422842209, 9.652035104710705371847621172671, 10.09865654224459165622898853511, 11.39114057626254254218486409575

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