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Properties

Label 2-336-112.109-c1-0-30
Degree $2$
Conductor $336$
Sign $-0.160 + 0.986i$
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  + (0.816 − 1.15i)2-s + (0.965 − 0.258i)3-s + (−0.665 − 1.88i)4-s + (0.397 + 0.106i)5-s + (0.490 − 1.32i)6-s + (0.990 − 2.45i)7-s + (−2.72 − 0.773i)8-s + (0.866 − 0.499i)9-s + (0.447 − 0.371i)10-s + (0.789 + 2.94i)11-s + (−1.13 − 1.64i)12-s + (−1.20 − 1.20i)13-s + (−2.02 − 3.14i)14-s + 0.411·15-s + (−3.11 + 2.50i)16-s + (−0.646 + 1.12i)17-s + ⋯
L(s)  = 1  + (0.577 − 0.816i)2-s + (0.557 − 0.149i)3-s + (−0.332 − 0.943i)4-s + (0.177 + 0.0476i)5-s + (0.200 − 0.541i)6-s + (0.374 − 0.927i)7-s + (−0.961 − 0.273i)8-s + (0.288 − 0.166i)9-s + (0.141 − 0.117i)10-s + (0.237 + 0.888i)11-s + (−0.326 − 0.476i)12-s + (−0.334 − 0.334i)13-s + (−0.540 − 0.841i)14-s + 0.106·15-s + (−0.778 + 0.627i)16-s + (−0.156 + 0.271i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.160 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.160 + 0.986i$
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: \((2,\ 336,\ (\ :1/2),\ -0.160 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34447 - 1.58149i\)
\(L(\frac12)\) \(\approx\) \(1.34447 - 1.58149i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.816 + 1.15i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (-0.990 + 2.45i)T \)
good5 \( 1 + (-0.397 - 0.106i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.789 - 2.94i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (1.20 + 1.20i)T + 13iT^{2} \)
17 \( 1 + (0.646 - 1.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.527 - 1.96i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-7.20 + 4.15i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.73 - 1.73i)T + 29iT^{2} \)
31 \( 1 + (2.95 - 5.11i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.97 - 1.60i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.372iT - 41T^{2} \)
43 \( 1 + (4.89 - 4.89i)T - 43iT^{2} \)
47 \( 1 + (-6.21 - 10.7i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.93 + 7.21i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.13 - 7.95i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.08 - 4.05i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-6.46 + 1.73i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.39iT - 71T^{2} \)
73 \( 1 + (6.66 + 3.84i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.20 + 10.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.59 - 7.59i)T + 83iT^{2} \)
89 \( 1 + (1.31 - 0.758i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−11.26803676462091978914910693726, −10.39688897016010359899572753954, −9.743604508356370008985375543595, −8.650440224122091586717165335079, −7.43221057890594254139773218319, −6.42523674629355695474002305320, −4.91377652699161879190098169997, −4.09367423850173385224590768368, −2.78411135227890510931107472173, −1.41002614314164773438850819132, 2.46980009987265326620820528611, 3.69215886078335154139084846276, 5.00402391620013063783786814900, 5.82480493778555844174766365204, 7.01700834894839791648701210152, 8.014896666825269921081831986269, 8.946951076984626414349697026623, 9.441282402810620534963966855147, 11.20543015795935618248527073326, 11.85025503136524720594795051305

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