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Properties

Label 2-336-112.109-c1-0-5
Degree $2$
Conductor $336$
Sign $0.104 - 0.994i$
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  + (−0.948 + 1.04i)2-s + (−0.965 + 0.258i)3-s + (−0.201 − 1.98i)4-s + (−2.71 − 0.726i)5-s + (0.644 − 1.25i)6-s + (−0.393 − 2.61i)7-s + (2.27 + 1.67i)8-s + (0.866 − 0.499i)9-s + (3.33 − 2.15i)10-s + (1.04 + 3.89i)11-s + (0.709 + 1.86i)12-s + (4.21 + 4.21i)13-s + (3.11 + 2.06i)14-s + 2.80·15-s + (−3.91 + 0.803i)16-s + (−2.10 + 3.64i)17-s + ⋯
L(s)  = 1  + (−0.670 + 0.741i)2-s + (−0.557 + 0.149i)3-s + (−0.100 − 0.994i)4-s + (−1.21 − 0.324i)5-s + (0.263 − 0.513i)6-s + (−0.148 − 0.988i)7-s + (0.805 + 0.592i)8-s + (0.288 − 0.166i)9-s + (1.05 − 0.681i)10-s + (0.314 + 1.17i)11-s + (0.204 + 0.539i)12-s + (1.16 + 1.16i)13-s + (0.833 + 0.552i)14-s + 0.724·15-s + (−0.979 + 0.200i)16-s + (−0.510 + 0.884i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.104 - 0.994i$
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: \((2,\ 336,\ (\ :1/2),\ 0.104 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.429029 + 0.386333i\)
\(L(\frac12)\) \(\approx\) \(0.429029 + 0.386333i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.948 - 1.04i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (0.393 + 2.61i)T \)
good5 \( 1 + (2.71 + 0.726i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.04 - 3.89i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-4.21 - 4.21i)T + 13iT^{2} \)
17 \( 1 + (2.10 - 3.64i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0315 - 0.117i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-6.12 + 3.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.932 - 0.932i)T + 29iT^{2} \)
31 \( 1 + (-0.233 + 0.404i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-10.5 - 2.81i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 7.38iT - 41T^{2} \)
43 \( 1 + (0.452 - 0.452i)T - 43iT^{2} \)
47 \( 1 + (0.826 + 1.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.86 - 10.6i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.903 + 3.37i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.62 - 6.05i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-6.79 + 1.81i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 0.947iT - 71T^{2} \)
73 \( 1 + (-8.20 - 4.73i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.03 + 1.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.86 + 3.86i)T + 83iT^{2} \)
89 \( 1 + (-3.38 + 1.95i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.22T + 97T^{2} \)
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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

−11.38214403522517974559594670734, −10.92168563861824185456735766468, −9.843083460200224051861940465712, −8.864400753485971524265863817751, −7.917414082930614416123845742420, −6.95388200114101634883608451830, −6.35465117808972961869481207564, −4.55310074619053012017218371444, −4.17631439062705193017774125972, −1.19601758284241715360058620423, 0.67861310261401413530390527980, 2.89361924280473856988349061638, 3.77258608053013964692011857067, 5.39541816369066640551102464372, 6.65862647620390569288203823518, 7.82353112927021845824596168716, 8.539133612861719019562498132877, 9.428871300872250501365842790548, 10.93605900347220175430320344945, 11.19338298106370532339269595014

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