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Properties

Label 2-304-304.197-c1-0-8
Degree $2$
Conductor $304$
Sign $0.542 - 0.839i$
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  + (1.38 − 0.285i)2-s + (−2.03 − 0.545i)3-s + (1.83 − 0.789i)4-s + (−1.08 + 4.05i)5-s + (−2.97 − 0.175i)6-s + 1.60i·7-s + (2.31 − 1.61i)8-s + (1.24 + 0.720i)9-s + (−0.348 + 5.92i)10-s + (1.21 + 1.21i)11-s + (−4.17 + 0.605i)12-s + (1.68 + 6.28i)13-s + (0.458 + 2.22i)14-s + (4.42 − 7.66i)15-s + (2.75 − 2.90i)16-s + (−0.878 − 1.52i)17-s + ⋯
L(s)  = 1  + (0.979 − 0.201i)2-s + (−1.17 − 0.314i)3-s + (0.918 − 0.394i)4-s + (−0.485 + 1.81i)5-s + (−1.21 − 0.0714i)6-s + 0.607i·7-s + (0.820 − 0.572i)8-s + (0.416 + 0.240i)9-s + (−0.110 + 1.87i)10-s + (0.367 + 0.367i)11-s + (−1.20 + 0.174i)12-s + (0.467 + 1.74i)13-s + (0.122 + 0.595i)14-s + (1.14 − 1.97i)15-s + (0.688 − 0.725i)16-s + (−0.213 − 0.369i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.542 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.542 - 0.839i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.542 - 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28524 + 0.699482i\)
\(L(\frac12)\) \(\approx\) \(1.28524 + 0.699482i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.285i)T \)
19 \( 1 + (-1.10 + 4.21i)T \)
good3 \( 1 + (2.03 + 0.545i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (1.08 - 4.05i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 - 1.60iT - 7T^{2} \)
11 \( 1 + (-1.21 - 1.21i)T + 11iT^{2} \)
13 \( 1 + (-1.68 - 6.28i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.878 + 1.52i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.604 + 0.348i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.09 - 4.08i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + 2.02T + 31T^{2} \)
37 \( 1 + (-1.71 - 1.71i)T + 37iT^{2} \)
41 \( 1 + (-0.726 + 0.419i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.90 + 10.8i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (2.23 - 3.87i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.86 - 1.83i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.361 - 1.34i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.55 + 9.52i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.149 - 0.557i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-9.37 + 5.41i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-13.2 + 7.64i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.20 + 3.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.69 - 2.69i)T - 83iT^{2} \)
89 \( 1 + (-5.92 - 3.42i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.68 - 11.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.74645766911121700302636771763, −11.21328318347616392107973438849, −10.68922760120048582478097449447, −9.297501514957217597624067101154, −7.29821689906852740848280776040, −6.67724193030256242374696509118, −6.17595242023905816090527164350, −4.81834486290365428817234664549, −3.54489807716240637263377704162, −2.21556076751141899209061581645, 0.964254357944911661498087875869, 3.70331962700997053296782452039, 4.58502021583768352620492236789, 5.50716666291053963903377557586, 6.05995942239576392891843373577, 7.76490867349602842086351798024, 8.382788469569484495425630710314, 10.00343946707091479642763374686, 11.01738014465889981649489337920, 11.70775885813338384917565620337

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