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Properties

Label 2-336-112.109-c1-0-12
Degree $2$
Conductor $336$
Sign $0.998 - 0.0582i$
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  + (−1.39 + 0.257i)2-s + (0.965 − 0.258i)3-s + (1.86 − 0.716i)4-s + (−0.588 − 0.157i)5-s + (−1.27 + 0.608i)6-s + (2.58 + 0.573i)7-s + (−2.41 + 1.47i)8-s + (0.866 − 0.499i)9-s + (0.859 + 0.0677i)10-s + (0.0594 + 0.221i)11-s + (1.61 − 1.17i)12-s + (1.86 + 1.86i)13-s + (−3.73 − 0.131i)14-s − 0.609·15-s + (2.97 − 2.67i)16-s + (1.70 − 2.96i)17-s + ⋯
L(s)  = 1  + (−0.983 + 0.182i)2-s + (0.557 − 0.149i)3-s + (0.933 − 0.358i)4-s + (−0.263 − 0.0705i)5-s + (−0.521 + 0.248i)6-s + (0.976 + 0.216i)7-s + (−0.852 + 0.522i)8-s + (0.288 − 0.166i)9-s + (0.271 + 0.0214i)10-s + (0.0179 + 0.0669i)11-s + (0.467 − 0.339i)12-s + (0.518 + 0.518i)13-s + (−0.999 − 0.0351i)14-s − 0.157·15-s + (0.743 − 0.668i)16-s + (0.414 − 0.717i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.998 - 0.0582i$
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: \((2,\ 336,\ (\ :1/2),\ 0.998 - 0.0582i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13668 + 0.0331481i\)
\(L(\frac12)\) \(\approx\) \(1.13668 + 0.0331481i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.257i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (-2.58 - 0.573i)T \)
good5 \( 1 + (0.588 + 0.157i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.0594 - 0.221i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.86 - 1.86i)T + 13iT^{2} \)
17 \( 1 + (-1.70 + 2.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.417 - 1.55i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.65 + 1.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.50 - 4.50i)T + 29iT^{2} \)
31 \( 1 + (-2.55 + 4.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.51 - 1.47i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 6.64iT - 41T^{2} \)
43 \( 1 + (-1.27 + 1.27i)T - 43iT^{2} \)
47 \( 1 + (3.46 + 6.00i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.876 - 3.27i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.87 + 7.01i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.93 - 7.21i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (14.2 - 3.81i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 3.66iT - 71T^{2} \)
73 \( 1 + (11.2 + 6.48i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.80 + 8.32i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.15 + 8.15i)T + 83iT^{2} \)
89 \( 1 + (10.9 - 6.33i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.7T + 97T^{2} \)
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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

−11.54259013231270784520342432487, −10.51365780222276277070422387565, −9.523257785688997265854741730610, −8.610395982953735369039399128063, −7.996677385373858005185341999031, −7.10924010609491264854218543295, −5.93571692804241952668449019204, −4.50554608777283785911405102639, −2.79526693313565644577329630332, −1.40561849368200620108259721514, 1.40348705436430101026785203497, 2.93079564177374195922398611339, 4.19734821865173159658180508205, 5.81592032139404050528897339335, 7.18974850365574252489637291076, 8.032501060387514647713639329837, 8.575338250597629834408636286137, 9.667549634144921108469556531563, 10.60716007005770652496907359033, 11.25016027726614865506648937807

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