Properties

Label 2-336-21.5-c3-0-19
Degree $2$
Conductor $336$
Sign $0.925 + 0.378i$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.60 − 4.94i)3-s + (−0.623 + 1.08i)5-s + (−10.0 + 15.5i)7-s + (−21.8 + 15.8i)9-s + (35.2 − 20.3i)11-s − 19.5i·13-s + (6.34 + 1.34i)15-s + (52.3 + 90.6i)17-s + (−35.0 − 20.2i)19-s + (92.9 + 24.8i)21-s + (69.6 + 40.2i)23-s + (61.7 + 106. i)25-s + (113. + 82.3i)27-s − 211. i·29-s + (86.6 − 50.0i)31-s + ⋯
L(s)  = 1  + (−0.309 − 0.950i)3-s + (−0.0557 + 0.0966i)5-s + (−0.544 + 0.838i)7-s + (−0.808 + 0.588i)9-s + (0.965 − 0.557i)11-s − 0.418i·13-s + (0.109 + 0.0231i)15-s + (0.746 + 1.29i)17-s + (−0.423 − 0.244i)19-s + (0.966 + 0.258i)21-s + (0.631 + 0.364i)23-s + (0.493 + 0.855i)25-s + (0.809 + 0.586i)27-s − 1.35i·29-s + (0.501 − 0.289i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.925 + 0.378i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.925 + 0.378i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.543650358\)
\(L(\frac12)\) \(\approx\) \(1.543650358\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.60 + 4.94i)T \)
7 \( 1 + (10.0 - 15.5i)T \)
good5 \( 1 + (0.623 - 1.08i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-35.2 + 20.3i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 19.5iT - 2.19e3T^{2} \)
17 \( 1 + (-52.3 - 90.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (35.0 + 20.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-69.6 - 40.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 211. iT - 2.43e4T^{2} \)
31 \( 1 + (-86.6 + 50.0i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-94.9 + 164. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 186.T + 6.89e4T^{2} \)
43 \( 1 + 158.T + 7.95e4T^{2} \)
47 \( 1 + (-179. + 310. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-366. + 211. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-312. - 541. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-699. - 403. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-149. - 258. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 455. iT - 3.57e5T^{2} \)
73 \( 1 + (434. - 250. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (30.9 - 53.6i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 73.1T + 5.71e5T^{2} \)
89 \( 1 + (57.3 - 99.3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.41e3iT - 9.12e5T^{2} \)
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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

−11.35620243161220968322656890590, −10.21864590925269550838100352562, −8.987931296795307447913706432510, −8.272073275588989481782033158774, −7.11699891957337068525490081514, −6.13431504455973465346614535321, −5.54807037378705516037836882200, −3.70157267120420125299584839621, −2.40589288554845625328020620534, −0.916672806243928519755480437737, 0.836447948393048858269632719189, 3.04781559496624860243982888139, 4.16223473831538819484836491619, 4.96275878641873398025903734299, 6.39505603327670653328869670524, 7.14110794363006248791000434552, 8.632836000377782658479251826272, 9.532363196576232772283183264323, 10.14391614637673299304494465659, 11.09489407473924266202679887408

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