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Properties

Label 2-336-112.109-c1-0-28
Degree $2$
Conductor $336$
Sign $0.0309 + 0.999i$
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  + (1.30 − 0.548i)2-s + (−0.965 + 0.258i)3-s + (1.39 − 1.43i)4-s + (−2.53 − 0.679i)5-s + (−1.11 + 0.867i)6-s + (2.36 − 1.18i)7-s + (1.03 − 2.63i)8-s + (0.866 − 0.499i)9-s + (−3.67 + 0.505i)10-s + (−1.08 − 4.06i)11-s + (−0.979 + 1.74i)12-s + (0.464 + 0.464i)13-s + (2.43 − 2.84i)14-s + 2.62·15-s + (−0.0927 − 3.99i)16-s + (3.46 − 5.99i)17-s + ⋯
L(s)  = 1  + (0.921 − 0.388i)2-s + (−0.557 + 0.149i)3-s + (0.698 − 0.715i)4-s + (−1.13 − 0.303i)5-s + (−0.455 + 0.354i)6-s + (0.894 − 0.447i)7-s + (0.366 − 0.930i)8-s + (0.288 − 0.166i)9-s + (−1.16 + 0.159i)10-s + (−0.328 − 1.22i)11-s + (−0.282 + 0.503i)12-s + (0.128 + 0.128i)13-s + (0.650 − 0.759i)14-s + 0.677·15-s + (−0.0231 − 0.999i)16-s + (0.839 − 1.45i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0309 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0309 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.0309 + 0.999i$
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: \((2,\ 336,\ (\ :1/2),\ 0.0309 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20647 - 1.16972i\)
\(L(\frac12)\) \(\approx\) \(1.20647 - 1.16972i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30 + 0.548i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (-2.36 + 1.18i)T \)
good5 \( 1 + (2.53 + 0.679i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.08 + 4.06i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-0.464 - 0.464i)T + 13iT^{2} \)
17 \( 1 + (-3.46 + 5.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.88 - 7.05i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.678 - 0.391i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.64 - 2.64i)T + 29iT^{2} \)
31 \( 1 + (1.15 - 1.99i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.13 - 2.44i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 4.12iT - 41T^{2} \)
43 \( 1 + (5.75 - 5.75i)T - 43iT^{2} \)
47 \( 1 + (-3.59 - 6.23i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.61 - 6.01i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.932 + 3.48i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.02 + 11.2i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-3.24 + 0.868i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 8.86iT - 71T^{2} \)
73 \( 1 + (2.81 + 1.62i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.01 - 6.94i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.1 + 11.1i)T + 83iT^{2} \)
89 \( 1 + (-1.40 + 0.808i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.91T + 97T^{2} \)
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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

−11.48881643688614669031359729516, −10.85808193609158626564081824675, −9.851239630847757398695551578573, −8.220434872186584760712817436357, −7.52573182969296220395972187074, −6.15274546442377275321049409940, −5.12992178993851585981911775130, −4.26410483355549045316527136737, −3.24074108254253501558638250098, −1.03926888785973427828043564258, 2.26012083750990610473625514269, 3.94715156484771061387211425657, 4.74766364996998776209201035091, 5.79276304650355495339595362926, 7.01807181659439113864040108456, 7.70750240283956218316071551655, 8.529599416716135666663275887881, 10.36026066395542588476257027943, 11.22462197696331911343477678009, 11.87796515705585636810978863613

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