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Properties

Label 2-336-112.109-c1-0-2
Degree $2$
Conductor $336$
Sign $-0.395 - 0.918i$
Analytic cond. $2.68297$
Root an. cond. $1.63797$
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  + (−0.978 − 1.02i)2-s + (−0.965 + 0.258i)3-s + (−0.0869 + 1.99i)4-s + (0.298 + 0.0800i)5-s + (1.20 + 0.733i)6-s + (−2.15 − 1.53i)7-s + (2.12 − 1.86i)8-s + (0.866 − 0.499i)9-s + (−0.210 − 0.383i)10-s + (1.29 + 4.84i)11-s + (−0.433 − 1.95i)12-s + (−4.35 − 4.35i)13-s + (0.531 + 3.70i)14-s − 0.309·15-s + (−3.98 − 0.347i)16-s + (−1.39 + 2.42i)17-s + ⋯
L(s)  = 1  + (−0.691 − 0.722i)2-s + (−0.557 + 0.149i)3-s + (−0.0434 + 0.999i)4-s + (0.133 + 0.0358i)5-s + (0.493 + 0.299i)6-s + (−0.813 − 0.581i)7-s + (0.751 − 0.659i)8-s + (0.288 − 0.166i)9-s + (−0.0665 − 0.121i)10-s + (0.391 + 1.45i)11-s + (−0.125 − 0.563i)12-s + (−1.20 − 1.20i)13-s + (0.142 + 0.989i)14-s − 0.0798·15-s + (−0.996 − 0.0868i)16-s + (−0.338 + 0.586i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.395 - 0.918i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.395 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.119016 + 0.180845i\)
\(L(\frac12)\) \(\approx\) \(0.119016 + 0.180845i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.978 + 1.02i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (2.15 + 1.53i)T \)
good5 \( 1 + (-0.298 - 0.0800i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.29 - 4.84i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (4.35 + 4.35i)T + 13iT^{2} \)
17 \( 1 + (1.39 - 2.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.71 - 6.38i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (5.30 - 3.06i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.26 - 2.26i)T + 29iT^{2} \)
31 \( 1 + (1.55 - 2.69i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.56 - 0.686i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 1.09iT - 41T^{2} \)
43 \( 1 + (3.86 - 3.86i)T - 43iT^{2} \)
47 \( 1 + (2.83 + 4.90i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.53 - 9.45i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.84 + 14.3i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.47 + 5.51i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (0.999 - 0.267i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.98iT - 71T^{2} \)
73 \( 1 + (9.78 + 5.64i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.66 + 13.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.57 - 8.57i)T + 83iT^{2} \)
89 \( 1 + (0.966 - 0.557i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.49T + 97T^{2} \)
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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

−12.00633237011711808775636320415, −10.54585629332318421754543804970, −10.04723310726666125777409408873, −9.607285449734189320225793190653, −8.034264226763080260869436090635, −7.26875601850241558984827903163, −6.16050348303251652772163510145, −4.58128011174220011468970053454, −3.53759895404744101447414785190, −1.89626741925123554985897113352, 0.19214942037388866762920831139, 2.37245930615513926193220122065, 4.47893757941424457075085727900, 5.72301637388369339954316704237, 6.44772845001023141252314472844, 7.22845813410985511414272554889, 8.604204166076064286775976047525, 9.303484018264570989403910166502, 10.07413734201247804707245503930, 11.35378301718213465814273337818

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