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Properties

Label 2-336-112.101-c2-0-51
Degree $2$
Conductor $336$
Sign $0.379 + 0.925i$
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  + (1.97 − 0.307i)2-s + (−1.67 + 0.448i)3-s + (3.81 − 1.21i)4-s + (−0.857 − 0.229i)5-s + (−3.16 + 1.39i)6-s + (−2.07 − 6.68i)7-s + (7.15 − 3.57i)8-s + (2.59 − 1.50i)9-s + (−1.76 − 0.190i)10-s + (8.56 − 2.29i)11-s + (−5.83 + 3.74i)12-s + (−3.60 − 3.60i)13-s + (−6.14 − 12.5i)14-s + 1.53·15-s + (13.0 − 9.25i)16-s + (−5.94 − 3.43i)17-s + ⋯
L(s)  = 1  + (0.988 − 0.153i)2-s + (−0.557 + 0.149i)3-s + (0.952 − 0.303i)4-s + (−0.171 − 0.0459i)5-s + (−0.528 + 0.233i)6-s + (−0.295 − 0.955i)7-s + (0.894 − 0.446i)8-s + (0.288 − 0.166i)9-s + (−0.176 − 0.0190i)10-s + (0.778 − 0.208i)11-s + (−0.485 + 0.311i)12-s + (−0.277 − 0.277i)13-s + (−0.439 − 0.898i)14-s + 0.102·15-s + (0.815 − 0.578i)16-s + (−0.349 − 0.201i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.379 + 0.925i$
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: \((2,\ 336,\ (\ :1),\ 0.379 + 0.925i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.04060 - 1.36815i\)
\(L(\frac12)\) \(\approx\) \(2.04060 - 1.36815i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.97 + 0.307i)T \)
3 \( 1 + (1.67 - 0.448i)T \)
7 \( 1 + (2.07 + 6.68i)T \)
good5 \( 1 + (0.857 + 0.229i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (-8.56 + 2.29i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (3.60 + 3.60i)T + 169iT^{2} \)
17 \( 1 + (5.94 + 3.43i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-3.67 + 13.7i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (-15.2 + 8.77i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-32.4 + 32.4i)T - 841iT^{2} \)
31 \( 1 + (-37.9 - 21.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (12.7 - 47.5i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 29.4T + 1.68e3T^{2} \)
43 \( 1 + (-23.0 - 23.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (37.2 - 21.4i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (19.9 - 5.35i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (0.868 + 3.24i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-16.9 + 63.3i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (-13.7 - 51.1i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 66.5iT - 5.04e3T^{2} \)
73 \( 1 + (45.0 - 78.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (0.339 + 0.587i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-31.3 - 31.3i)T + 6.88e3iT^{2} \)
89 \( 1 + (40.0 + 69.3i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 132. iT - 9.40e3T^{2} \)
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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

−11.42020291353707860095072414580, −10.42221704791234317463291646028, −9.726515310730585476475531087882, −8.105943249649318235177587250150, −6.83089986109120551288186168129, −6.37703347290534931964453044636, −4.91186488875245690533859749021, −4.20355322350714414543670593434, −2.94993498714959777598849808262, −0.949269903159245784536971503736, 1.86255169000539030721988166113, 3.34875447776891274410519793743, 4.59177070040168076898207713721, 5.63864537192870817946477098232, 6.45279537737888498961187175455, 7.33350985443547187794398151817, 8.591676422907625225886920153092, 9.787817848319915051650949858920, 10.94689868992700416992203330289, 11.91725998099613635502492602478

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