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Properties

Label 2-336-112.101-c2-0-50
Degree $2$
Conductor $336$
Sign $0.387 + 0.921i$
Analytic cond. $9.15533$
Root an. cond. $3.02577$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.371 + 1.96i)2-s + (−1.67 + 0.448i)3-s + (−3.72 + 1.45i)4-s + (5.47 + 1.46i)5-s + (−1.50 − 3.12i)6-s + (−1.24 − 6.88i)7-s + (−4.25 − 6.77i)8-s + (2.59 − 1.50i)9-s + (−0.849 + 11.2i)10-s + (−16.2 + 4.36i)11-s + (5.57 − 4.11i)12-s + (−17.7 − 17.7i)13-s + (13.0 − 5.00i)14-s − 9.81·15-s + (11.7 − 10.8i)16-s + (4.91 + 2.83i)17-s + ⋯
L(s)  = 1  + (0.185 + 0.982i)2-s + (−0.557 + 0.149i)3-s + (−0.931 + 0.364i)4-s + (1.09 + 0.293i)5-s + (−0.250 − 0.520i)6-s + (−0.177 − 0.984i)7-s + (−0.531 − 0.847i)8-s + (0.288 − 0.166i)9-s + (−0.0849 + 1.12i)10-s + (−1.48 + 0.396i)11-s + (0.464 − 0.342i)12-s + (−1.36 − 1.36i)13-s + (0.933 − 0.357i)14-s − 0.654·15-s + (0.733 − 0.679i)16-s + (0.289 + 0.166i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.387 + 0.921i$
Analytic conductor: \(9.15533\)
Root analytic conductor: \(3.02577\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1),\ 0.387 + 0.921i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.455694 - 0.302788i\)
\(L(\frac12)\) \(\approx\) \(0.455694 - 0.302788i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.371 - 1.96i)T \)
3 \( 1 + (1.67 - 0.448i)T \)
7 \( 1 + (1.24 + 6.88i)T \)
good5 \( 1 + (-5.47 - 1.46i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (16.2 - 4.36i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (17.7 + 17.7i)T + 169iT^{2} \)
17 \( 1 + (-4.91 - 2.83i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-3.96 + 14.7i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (29.4 - 17.0i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-18.7 + 18.7i)T - 841iT^{2} \)
31 \( 1 + (12.9 + 7.50i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-10.7 + 40.1i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 73.7T + 1.68e3T^{2} \)
43 \( 1 + (-21.1 - 21.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (23.1 - 13.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (66.9 - 17.9i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (18.9 + 70.8i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (16.4 - 61.4i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (9.66 + 36.0i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 4.09iT - 5.04e3T^{2} \)
73 \( 1 + (66.4 - 115. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (20.4 + 35.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (25.0 + 25.0i)T + 6.88e3iT^{2} \)
89 \( 1 + (2.52 + 4.38i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 29.6iT - 9.40e3T^{2} \)
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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

−10.85446218725749106212467261607, −9.901514229333444963170877974156, −9.745258161428232283483758038383, −7.76019890434894865173889546498, −7.46485733012939119052468574945, −6.08817799921183752704255940403, −5.43407849960647289438014924508, −4.46170486137045848844310261662, −2.76617198856635098781485874551, −0.24043176443045805397162878294, 1.85813844365798037150463676358, 2.70739713540560385082127908916, 4.66359904915433771988261464007, 5.45334352355789208977879211205, 6.20513968884217283551791785667, 7.919508402141457215067103163522, 9.118674702023965943908711473083, 9.840129920206871296078988316922, 10.49846974134554267394427169702, 11.71190326345507838240097317414

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