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Properties

Label 2-304-16.5-c1-0-1
Degree $2$
Conductor $304$
Sign $-0.988 - 0.150i$
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 + 0.359i)2-s + (−1.75 + 1.75i)3-s + (1.74 − 0.982i)4-s + (0.519 + 0.519i)5-s + (1.76 − 3.03i)6-s + 1.79i·7-s + (−2.02 + 1.96i)8-s − 3.15i·9-s + (−0.896 − 0.523i)10-s + (3.97 + 3.97i)11-s + (−1.33 + 4.78i)12-s + (0.114 − 0.114i)13-s + (−0.644 − 2.45i)14-s − 1.82·15-s + (2.06 − 3.42i)16-s − 5.90·17-s + ⋯
L(s)  = 1  + (−0.967 + 0.253i)2-s + (−1.01 + 1.01i)3-s + (0.870 − 0.491i)4-s + (0.232 + 0.232i)5-s + (0.722 − 1.23i)6-s + 0.678i·7-s + (−0.717 + 0.696i)8-s − 1.05i·9-s + (−0.283 − 0.165i)10-s + (1.19 + 1.19i)11-s + (−0.384 + 1.38i)12-s + (0.0318 − 0.0318i)13-s + (−0.172 − 0.656i)14-s − 0.470·15-s + (0.517 − 0.855i)16-s − 1.43·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.988 - 0.150i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ -0.988 - 0.150i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0360437 + 0.476655i\)
\(L(\frac12)\) \(\approx\) \(0.0360437 + 0.476655i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.359i)T \)
19 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (1.75 - 1.75i)T - 3iT^{2} \)
5 \( 1 + (-0.519 - 0.519i)T + 5iT^{2} \)
7 \( 1 - 1.79iT - 7T^{2} \)
11 \( 1 + (-3.97 - 3.97i)T + 11iT^{2} \)
13 \( 1 + (-0.114 + 0.114i)T - 13iT^{2} \)
17 \( 1 + 5.90T + 17T^{2} \)
23 \( 1 - 2.02iT - 23T^{2} \)
29 \( 1 + (2.66 - 2.66i)T - 29iT^{2} \)
31 \( 1 + 7.42T + 31T^{2} \)
37 \( 1 + (3.31 + 3.31i)T + 37iT^{2} \)
41 \( 1 - 1.52iT - 41T^{2} \)
43 \( 1 + (-4.08 - 4.08i)T + 43iT^{2} \)
47 \( 1 + 7.56T + 47T^{2} \)
53 \( 1 + (-8.64 - 8.64i)T + 53iT^{2} \)
59 \( 1 + (-3.11 - 3.11i)T + 59iT^{2} \)
61 \( 1 + (-9.26 + 9.26i)T - 61iT^{2} \)
67 \( 1 + (1.50 - 1.50i)T - 67iT^{2} \)
71 \( 1 - 2.02iT - 71T^{2} \)
73 \( 1 - 9.15iT - 73T^{2} \)
79 \( 1 + 7.21T + 79T^{2} \)
83 \( 1 + (0.619 - 0.619i)T - 83iT^{2} \)
89 \( 1 + 1.20iT - 89T^{2} \)
97 \( 1 - 7.23T + 97T^{2} \)
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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

−11.70314143304994655528868354805, −11.09402445085734634804078620338, −10.20694602727632483189907968579, −9.427018673485310027198775366119, −8.788825179336026198913186398377, −7.16770143711663822780172019491, −6.32431056063175063324722050831, −5.38758284174613405947418252226, −4.16571946510479724495271663978, −2.06966620426000296579833065853, 0.53414350071153116667015234650, 1.79271999358138251506779568302, 3.79062970820635127306221574329, 5.70351923627167707458340987295, 6.65748796380226891433384507718, 7.17430527884158209338703060823, 8.523809766101816386230371291214, 9.265228919934872488834284663615, 10.64392380640399045257390245196, 11.29849564396000476965187484430

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