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Properties

Label 2-304-19.6-c1-0-2
Degree $2$
Conductor $304$
Sign $0.500 - 0.865i$
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  + (1.76 + 0.642i)3-s + (0.0603 + 0.342i)5-s + (−2.37 + 4.12i)7-s + (0.407 + 0.342i)9-s + (2.37 + 4.12i)11-s + (1.76 − 0.642i)13-s + (−0.113 + 0.642i)15-s + (1.17 − 0.984i)17-s + (0.694 − 4.30i)19-s + (−6.85 + 5.74i)21-s + (1.06 − 6.01i)23-s + (4.58 − 1.66i)25-s + (−2.31 − 4.01i)27-s + (−1.52 − 1.27i)29-s + (−1.68 + 2.91i)31-s + ⋯
L(s)  = 1  + (1.01 + 0.371i)3-s + (0.0269 + 0.152i)5-s + (−0.899 + 1.55i)7-s + (0.135 + 0.114i)9-s + (0.717 + 1.24i)11-s + (0.489 − 0.178i)13-s + (−0.0292 + 0.165i)15-s + (0.284 − 0.238i)17-s + (0.159 − 0.987i)19-s + (−1.49 + 1.25i)21-s + (0.221 − 1.25i)23-s + (0.917 − 0.333i)25-s + (−0.446 − 0.773i)27-s + (−0.282 − 0.236i)29-s + (−0.302 + 0.524i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.500 - 0.865i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.500 - 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43949 + 0.830467i\)
\(L(\frac12)\) \(\approx\) \(1.43949 + 0.830467i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.694 + 4.30i)T \)
good3 \( 1 + (-1.76 - 0.642i)T + (2.29 + 1.92i)T^{2} \)
5 \( 1 + (-0.0603 - 0.342i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (2.37 - 4.12i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.37 - 4.12i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.76 + 0.642i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-1.17 + 0.984i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-1.06 + 6.01i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (1.52 + 1.27i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (1.68 - 2.91i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.38T + 37T^{2} \)
41 \( 1 + (-4.21 - 1.53i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.365 + 2.07i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-3.76 - 3.16i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (2.18 - 12.3i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (4.33 - 3.63i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-1.73 + 9.84i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (8.11 + 6.81i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.99 + 11.3i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-13.7 - 4.99i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-2.21 - 0.807i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (5.76 - 9.99i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.29 - 1.92i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-5.86 + 4.92i)T + (16.8 - 95.5i)T^{2} \)
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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

−12.22695358127299818701095693726, −10.83501238693572551490197986822, −9.544160158078355965461326923411, −9.190172587195317496947859141981, −8.421419120353931704177160716795, −6.98515161506368637783206178816, −6.04400807128333806241180227752, −4.63205214210317137697661452602, −3.20835474087008706022776194123, −2.39748348811973948071794164679, 1.27028711462027867839972661674, 3.34011484307727595897162006318, 3.78553708729769592232839687256, 5.74906788041497234876407478615, 6.89646273695139038193372356822, 7.73625805442606153907009343357, 8.706880165486976551712169226499, 9.536565052570111493148901947049, 10.59427049764523268842594482153, 11.46231849583872245462374239472

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