Properties

Label 2-304-304.3-c1-0-2
Degree $2$
Conductor $304$
Sign $-0.254 - 0.966i$
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  + (−1.36 − 0.351i)2-s + (−1.52 − 0.712i)3-s + (1.75 + 0.961i)4-s + (2.13 + 3.04i)5-s + (1.84 + 1.51i)6-s + (0.591 + 1.02i)7-s + (−2.06 − 1.93i)8-s + (−0.0997 − 0.118i)9-s + (−1.85 − 4.91i)10-s + (−6.12 + 1.64i)11-s + (−1.99 − 2.72i)12-s + (0.768 − 0.358i)13-s + (−0.450 − 1.61i)14-s + (−1.08 − 6.17i)15-s + (2.14 + 3.37i)16-s + (0.0293 + 0.0246i)17-s + ⋯
L(s)  = 1  + (−0.968 − 0.248i)2-s + (−0.882 − 0.411i)3-s + (0.876 + 0.480i)4-s + (0.952 + 1.36i)5-s + (0.752 + 0.617i)6-s + (0.223 + 0.387i)7-s + (−0.729 − 0.683i)8-s + (−0.0332 − 0.0396i)9-s + (−0.585 − 1.55i)10-s + (−1.84 + 0.494i)11-s + (−0.575 − 0.785i)12-s + (0.213 − 0.0993i)13-s + (−0.120 − 0.430i)14-s + (−0.280 − 1.59i)15-s + (0.537 + 0.843i)16-s + (0.00711 + 0.00596i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.254 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.254 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.254 - 0.966i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ -0.254 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.282424 + 0.366494i\)
\(L(\frac12)\) \(\approx\) \(0.282424 + 0.366494i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.351i)T \)
19 \( 1 + (3.35 - 2.78i)T \)
good3 \( 1 + (1.52 + 0.712i)T + (1.92 + 2.29i)T^{2} \)
5 \( 1 + (-2.13 - 3.04i)T + (-1.71 + 4.69i)T^{2} \)
7 \( 1 + (-0.591 - 1.02i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (6.12 - 1.64i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.768 + 0.358i)T + (8.35 - 9.95i)T^{2} \)
17 \( 1 + (-0.0293 - 0.0246i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.511 - 2.90i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (0.147 - 1.68i)T + (-28.5 - 5.03i)T^{2} \)
31 \( 1 + (2.04 + 3.53i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.10 - 4.10i)T + 37iT^{2} \)
41 \( 1 + (11.1 - 4.05i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-5.24 + 3.67i)T + (14.7 - 40.4i)T^{2} \)
47 \( 1 + (-2.81 - 3.35i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (3.60 - 5.15i)T + (-18.1 - 49.8i)T^{2} \)
59 \( 1 + (1.05 - 0.0920i)T + (58.1 - 10.2i)T^{2} \)
61 \( 1 + (7.22 - 10.3i)T + (-20.8 - 57.3i)T^{2} \)
67 \( 1 + (-0.417 - 0.0364i)T + (65.9 + 11.6i)T^{2} \)
71 \( 1 + (-11.8 - 2.09i)T + (66.7 + 24.2i)T^{2} \)
73 \( 1 + (1.51 + 4.16i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (-10.3 + 3.77i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-11.1 - 2.99i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (5.37 + 1.95i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-8.28 + 9.87i)T + (-16.8 - 95.5i)T^{2} \)
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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

−11.70878013042279703761888826252, −10.72974020353293137185413994142, −10.45445297217196331184675767209, −9.430233011401666970713736707184, −8.061124906087305807677211478572, −7.14836084325586843596767875824, −6.22511159800383423229046144572, −5.50011252072762915121807438355, −3.02199607927282901616492203948, −1.99534163868324915095643466408, 0.47612557140666620014600794422, 2.26668429144273972079374714487, 4.88359438245629292110922948224, 5.41113358964253101674457903385, 6.36315238689458903216963434272, 7.917258976480960218575417030338, 8.618689319247546409901875752131, 9.604621756575368699319928575857, 10.65460024203125099331352032897, 10.87477965821479549282042435549

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