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Properties

Label 2-304-19.4-c1-0-4
Degree $2$
Conductor $304$
Sign $0.433 + 0.901i$
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  + (−0.140 − 0.797i)3-s + (−2.64 + 2.21i)5-s + (2.01 − 3.48i)7-s + (2.20 − 0.802i)9-s + (−1.26 − 2.18i)11-s + (0.892 − 5.06i)13-s + (2.13 + 1.79i)15-s + (4.64 + 1.69i)17-s + (4.26 + 0.911i)19-s + (−3.06 − 1.11i)21-s + (−4.08 − 3.42i)23-s + (1.19 − 6.77i)25-s + (−2.16 − 3.74i)27-s + (−5.48 + 1.99i)29-s + (−2.69 + 4.66i)31-s + ⋯
L(s)  = 1  + (−0.0811 − 0.460i)3-s + (−1.18 + 0.990i)5-s + (0.760 − 1.31i)7-s + (0.734 − 0.267i)9-s + (−0.380 − 0.659i)11-s + (0.247 − 1.40i)13-s + (0.551 + 0.462i)15-s + (1.12 + 0.410i)17-s + (0.977 + 0.209i)19-s + (−0.668 − 0.243i)21-s + (−0.851 − 0.714i)23-s + (0.238 − 1.35i)25-s + (−0.416 − 0.720i)27-s + (−1.01 + 0.370i)29-s + (−0.483 + 0.837i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.433 + 0.901i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.433 + 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.959125 - 0.602889i\)
\(L(\frac12)\) \(\approx\) \(0.959125 - 0.602889i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.26 - 0.911i)T \)
good3 \( 1 + (0.140 + 0.797i)T + (-2.81 + 1.02i)T^{2} \)
5 \( 1 + (2.64 - 2.21i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-2.01 + 3.48i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.26 + 2.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.892 + 5.06i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-4.64 - 1.69i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (4.08 + 3.42i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (5.48 - 1.99i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (2.69 - 4.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.23T + 37T^{2} \)
41 \( 1 + (0.387 + 2.19i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (1.92 - 1.61i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-0.138 + 0.0503i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-5.51 - 4.62i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-4.95 - 1.80i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-1.60 - 1.34i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (4.67 - 1.70i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (3.74 - 3.14i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-1.76 - 9.99i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-1.22 - 6.95i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-5.85 + 10.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.164 - 0.930i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (11.2 + 4.09i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.45712739805824027219566547236, −10.61855207412761362975069501544, −10.15830466254390451124856704673, −8.137778804056319009281706605106, −7.66929542429796723253035167729, −7.05811925040079262515740769567, −5.64113449960284133800163669756, −4.04245920100083529765881559777, −3.28151265443998202836518341621, −0.950747528041739606521504069890, 1.81406361830751888865478757119, 3.86649323739808077848072767432, 4.77472068098841459383355179575, 5.53695860293103891490353895828, 7.42414854762539268482908690054, 8.011051439548662055493180382084, 9.167149883934790182553605662541, 9.729573217247846716775748877065, 11.38593019477513844378167001000, 11.76920282659465555059121555747

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