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Properties

Label 2-336-112.109-c1-0-21
Degree $2$
Conductor $336$
Sign $0.871 + 0.489i$
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  + (−0.244 + 1.39i)2-s + (0.965 − 0.258i)3-s + (−1.88 − 0.681i)4-s + (−3.28 − 0.881i)5-s + (0.124 + 1.40i)6-s + (2.12 − 1.57i)7-s + (1.40 − 2.45i)8-s + (0.866 − 0.499i)9-s + (2.03 − 4.36i)10-s + (−1.25 − 4.68i)11-s + (−1.99 − 0.171i)12-s + (1.85 + 1.85i)13-s + (1.67 + 3.34i)14-s − 3.40·15-s + (3.07 + 2.56i)16-s + (1.09 − 1.88i)17-s + ⋯
L(s)  = 1  + (−0.172 + 0.984i)2-s + (0.557 − 0.149i)3-s + (−0.940 − 0.340i)4-s + (−1.47 − 0.394i)5-s + (0.0507 + 0.575i)6-s + (0.802 − 0.596i)7-s + (0.497 − 0.867i)8-s + (0.288 − 0.166i)9-s + (0.642 − 1.38i)10-s + (−0.378 − 1.41i)11-s + (−0.575 − 0.0494i)12-s + (0.514 + 0.514i)13-s + (0.448 + 0.893i)14-s − 0.879·15-s + (0.768 + 0.640i)16-s + (0.264 − 0.458i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.871 + 0.489i$
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: \((2,\ 336,\ (\ :1/2),\ 0.871 + 0.489i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.973764 - 0.254727i\)
\(L(\frac12)\) \(\approx\) \(0.973764 - 0.254727i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.244 - 1.39i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (-2.12 + 1.57i)T \)
good5 \( 1 + (3.28 + 0.881i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.25 + 4.68i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.85 - 1.85i)T + 13iT^{2} \)
17 \( 1 + (-1.09 + 1.88i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.974 + 3.63i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.98 + 1.14i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.15 + 5.15i)T + 29iT^{2} \)
31 \( 1 + (4.18 - 7.24i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.63 + 1.50i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 5.68iT - 41T^{2} \)
43 \( 1 + (-6.30 + 6.30i)T - 43iT^{2} \)
47 \( 1 + (-4.68 - 8.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.35 - 5.05i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.88 - 10.7i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.798 + 2.98i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (5.28 - 1.41i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 3.87iT - 71T^{2} \)
73 \( 1 + (-5.68 - 3.28i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.37 + 12.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.64 - 8.64i)T + 83iT^{2} \)
89 \( 1 + (8.73 - 5.04i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.63T + 97T^{2} \)
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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

−11.35478890797249429387080823998, −10.66399240413393031115568286663, −8.931081146334988743367786114833, −8.627313974766571898940938917653, −7.60514509928822806892218202367, −7.16422066586721420754184909603, −5.54783944655337289525132632045, −4.39252760783613342399635104077, −3.54052886865327812763173838528, −0.76584298339902537251761746607, 1.89297308305234881432713767513, 3.31729230610522794081016974522, 4.16831543606711472319842580533, 5.27309788121777750398770450966, 7.48437801001972667564017459620, 7.915268105681016522833063264773, 8.815159387139339388925122944223, 9.908105535174266040449525352059, 10.85775098934660334157129620430, 11.53813942680934824143986124749

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