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Properties

Label 2-336-112.101-c2-0-38
Degree $2$
Conductor $336$
Sign $0.489 + 0.872i$
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  + (1.39 − 1.43i)2-s + (−1.67 + 0.448i)3-s + (−0.118 − 3.99i)4-s + (2.73 + 0.733i)5-s + (−1.68 + 3.02i)6-s + (5.81 + 3.90i)7-s + (−5.90 − 5.40i)8-s + (2.59 − 1.50i)9-s + (4.86 − 2.90i)10-s + (6.89 − 1.84i)11-s + (1.99 + 6.63i)12-s + (15.7 + 15.7i)13-s + (13.6 − 2.90i)14-s − 4.91·15-s + (−15.9 + 0.946i)16-s + (−12.8 − 7.44i)17-s + ⋯
L(s)  = 1  + (0.696 − 0.717i)2-s + (−0.557 + 0.149i)3-s + (−0.0295 − 0.999i)4-s + (0.547 + 0.146i)5-s + (−0.281 + 0.504i)6-s + (0.830 + 0.557i)7-s + (−0.737 − 0.675i)8-s + (0.288 − 0.166i)9-s + (0.486 − 0.290i)10-s + (0.626 − 0.167i)11-s + (0.165 + 0.553i)12-s + (1.21 + 1.21i)13-s + (0.978 − 0.207i)14-s − 0.327·15-s + (−0.998 + 0.0591i)16-s + (−0.758 − 0.437i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.872i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.489 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.489 + 0.872i$
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: \((2,\ 336,\ (\ :1),\ 0.489 + 0.872i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.12450 - 1.24427i\)
\(L(\frac12)\) \(\approx\) \(2.12450 - 1.24427i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 1.43i)T \)
3 \( 1 + (1.67 - 0.448i)T \)
7 \( 1 + (-5.81 - 3.90i)T \)
good5 \( 1 + (-2.73 - 0.733i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (-6.89 + 1.84i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (-15.7 - 15.7i)T + 169iT^{2} \)
17 \( 1 + (12.8 + 7.44i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-8.66 + 32.3i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (-18.7 + 10.8i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (7.54 - 7.54i)T - 841iT^{2} \)
31 \( 1 + (-34.0 - 19.6i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-5.60 + 20.9i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 53.6T + 1.68e3T^{2} \)
43 \( 1 + (26.3 + 26.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (-4.19 + 2.42i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (85.6 - 22.9i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (18.4 + 68.6i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (29.2 - 108. i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (-13.6 - 50.8i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 28.5iT - 5.04e3T^{2} \)
73 \( 1 + (9.86 - 17.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-40.5 - 70.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (110. + 110. i)T + 6.88e3iT^{2} \)
89 \( 1 + (-42.3 - 73.4i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 45.7iT - 9.40e3T^{2} \)
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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

−11.36607970640037343591860554503, −10.69284256447312953201531483016, −9.295696433481549089499635842604, −8.890049184417940421195651338609, −6.83835644260262400845091641938, −6.14105411888775160191956141151, −5.01335937745415337511960341201, −4.21457156618495270621163492726, −2.56746260648390487874235474591, −1.25740678118100495288967525575, 1.47383611312934017114052510581, 3.54977062504074892426095876033, 4.63378215070077172376085349014, 5.76576307566220230079138289281, 6.31402514774090992464501671609, 7.67311519111707925718420919619, 8.260686645530189774613632514545, 9.617013897351676002715140314388, 10.85104018019219176606680516017, 11.53717824842076761257440923032

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