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Properties

Label 2-336-112.109-c1-0-27
Degree $2$
Conductor $336$
Sign $-0.253 + 0.967i$
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  + (−1.17 + 0.781i)2-s + (0.965 − 0.258i)3-s + (0.778 − 1.84i)4-s + (−1.63 − 0.436i)5-s + (−0.936 + 1.05i)6-s + (−2.22 − 1.43i)7-s + (0.522 + 2.77i)8-s + (0.866 − 0.499i)9-s + (2.26 − 0.759i)10-s + (−0.0320 − 0.119i)11-s + (0.275 − 1.98i)12-s + (−3.88 − 3.88i)13-s + (3.74 − 0.0522i)14-s − 1.68·15-s + (−2.78 − 2.86i)16-s + (0.0515 − 0.0893i)17-s + ⋯
L(s)  = 1  + (−0.833 + 0.552i)2-s + (0.557 − 0.149i)3-s + (0.389 − 0.921i)4-s + (−0.729 − 0.195i)5-s + (−0.382 + 0.432i)6-s + (−0.841 − 0.540i)7-s + (0.184 + 0.982i)8-s + (0.288 − 0.166i)9-s + (0.715 − 0.240i)10-s + (−0.00965 − 0.0360i)11-s + (0.0794 − 0.571i)12-s + (−1.07 − 1.07i)13-s + (0.999 − 0.0139i)14-s − 0.435·15-s + (−0.696 − 0.717i)16-s + (0.0125 − 0.0216i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.253 + 0.967i$
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: \((2,\ 336,\ (\ :1/2),\ -0.253 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.314541 - 0.407673i\)
\(L(\frac12)\) \(\approx\) \(0.314541 - 0.407673i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 - 0.781i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (2.22 + 1.43i)T \)
good5 \( 1 + (1.63 + 0.436i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.0320 + 0.119i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (3.88 + 3.88i)T + 13iT^{2} \)
17 \( 1 + (-0.0515 + 0.0893i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.855 + 3.19i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (5.38 - 3.10i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.66 + 2.66i)T + 29iT^{2} \)
31 \( 1 + (-2.17 + 3.76i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.88 - 2.38i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 7.82iT - 41T^{2} \)
43 \( 1 + (0.793 - 0.793i)T - 43iT^{2} \)
47 \( 1 + (-1.78 - 3.08i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.15 + 11.7i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.76 - 6.58i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.64 + 6.12i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (3.39 - 0.908i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 + (-8.37 - 4.83i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.10 - 12.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.85 - 3.85i)T + 83iT^{2} \)
89 \( 1 + (-2.98 + 1.72i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 16.7T + 97T^{2} \)
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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

−11.11654833232754711033864159268, −9.879011300754581249056586883158, −9.598151348086127123004804284449, −8.169709532829638269128089980823, −7.70081867754330292000691087156, −6.81380735157246501929045735119, −5.57393305577031979228666195993, −4.08778569491083117970398507061, −2.59637824885014918848417401776, −0.41915765917494837965737860325, 2.16976067262633560955973226963, 3.31267824392349835159288587721, 4.35311925273902659189373226748, 6.33634470312069191140329519178, 7.39816295406353622521248757066, 8.150573639193987424345279807771, 9.269077441224664193274846503615, 9.735395940402097994005745804095, 10.76480014638115038466853392142, 11.99545844304724053144451053731

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