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Properties

Label 2-336-112.101-c2-0-5
Degree $2$
Conductor $336$
Sign $-0.964 + 0.263i$
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  + (−1.08 + 1.67i)2-s + (−1.67 + 0.448i)3-s + (−1.62 − 3.65i)4-s + (4.74 + 1.27i)5-s + (1.06 − 3.29i)6-s + (−6.99 − 0.313i)7-s + (7.90 + 1.24i)8-s + (2.59 − 1.50i)9-s + (−7.30 + 6.57i)10-s + (−0.461 + 0.123i)11-s + (4.36 + 5.38i)12-s + (5.87 + 5.87i)13-s + (8.13 − 11.3i)14-s − 8.51·15-s + (−10.6 + 11.9i)16-s + (−3.44 − 1.98i)17-s + ⋯
L(s)  = 1  + (−0.544 + 0.838i)2-s + (−0.557 + 0.149i)3-s + (−0.407 − 0.913i)4-s + (0.949 + 0.254i)5-s + (0.178 − 0.549i)6-s + (−0.998 − 0.0448i)7-s + (0.987 + 0.155i)8-s + (0.288 − 0.166i)9-s + (−0.730 + 0.657i)10-s + (−0.0419 + 0.0112i)11-s + (0.363 + 0.448i)12-s + (0.451 + 0.451i)13-s + (0.581 − 0.813i)14-s − 0.567·15-s + (−0.668 + 0.744i)16-s + (−0.202 − 0.117i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.964 + 0.263i$
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: \((2,\ 336,\ (\ :1),\ -0.964 + 0.263i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0565054 - 0.421438i\)
\(L(\frac12)\) \(\approx\) \(0.0565054 - 0.421438i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.08 - 1.67i)T \)
3 \( 1 + (1.67 - 0.448i)T \)
7 \( 1 + (6.99 + 0.313i)T \)
good5 \( 1 + (-4.74 - 1.27i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (0.461 - 0.123i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (-5.87 - 5.87i)T + 169iT^{2} \)
17 \( 1 + (3.44 + 1.98i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (6.74 - 25.1i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (22.3 - 12.8i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (1.30 - 1.30i)T - 841iT^{2} \)
31 \( 1 + (7.89 + 4.55i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (2.45 - 9.14i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 36.6T + 1.68e3T^{2} \)
43 \( 1 + (44.4 + 44.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (29.6 - 17.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (64.8 - 17.3i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (9.62 + 35.9i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-5.04 + 18.8i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (-27.3 - 102. i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 36.6iT - 5.04e3T^{2} \)
73 \( 1 + (35.8 - 62.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-31.4 - 54.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (1.88 + 1.88i)T + 6.88e3iT^{2} \)
89 \( 1 + (3.53 + 6.12i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 154. iT - 9.40e3T^{2} \)
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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

−11.69587103601698877367811188017, −10.43541747148604647278962454516, −9.966064720271393873427020543681, −9.220500654146549959476761458712, −8.034941644814691837231171222009, −6.71816920761182460027430158182, −6.19715600765741377369749071236, −5.38782748465302029696921609474, −3.86080453322760330345678751282, −1.76558800807409013141571316672, 0.24614994399923229704031693153, 1.88122160736448093586525969126, 3.21672755522073773368133646352, 4.69537108511640927848980350359, 5.99076114000923411390473970206, 6.89300143060225295617622627130, 8.278458116122855410225975815290, 9.263649563521522655094809790012, 9.966486651464769694344045952261, 10.71493758108704465177916274068

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