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Properties

Label 2-336-112.109-c1-0-17
Degree $2$
Conductor $336$
Sign $0.997 + 0.0684i$
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  + (−1.22 + 0.707i)2-s + (−0.965 + 0.258i)3-s + (0.999 − 1.73i)4-s + (3.69 + 0.990i)5-s + (0.999 − i)6-s + (0.358 − 2.62i)7-s + 2.82i·8-s + (0.866 − 0.499i)9-s + (−5.22 + 1.40i)10-s + (−0.624 − 2.33i)11-s + (−0.517 + 1.93i)12-s + (−2.41 − 2.41i)13-s + (1.41 + 3.46i)14-s − 3.82·15-s + (−2.00 − 3.46i)16-s + (3.41 − 5.91i)17-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)2-s + (−0.557 + 0.149i)3-s + (0.499 − 0.866i)4-s + (1.65 + 0.443i)5-s + (0.408 − 0.408i)6-s + (0.135 − 0.990i)7-s + 0.999i·8-s + (0.288 − 0.166i)9-s + (−1.65 + 0.443i)10-s + (−0.188 − 0.703i)11-s + (−0.149 + 0.557i)12-s + (−0.669 − 0.669i)13-s + (0.377 + 0.925i)14-s − 0.988·15-s + (−0.500 − 0.866i)16-s + (0.828 − 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.997 + 0.0684i$
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: \((2,\ 336,\ (\ :1/2),\ 0.997 + 0.0684i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.975553 - 0.0334148i\)
\(L(\frac12)\) \(\approx\) \(0.975553 - 0.0334148i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.707i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (-0.358 + 2.62i)T \)
good5 \( 1 + (-3.69 - 0.990i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.624 + 2.33i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (2.41 + 2.41i)T + 13iT^{2} \)
17 \( 1 + (-3.41 + 5.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.732 - 2.73i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.210 + 0.121i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.12 - 6.12i)T + 29iT^{2} \)
31 \( 1 + (0.207 - 0.358i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.73 + 0.732i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 3.41iT - 41T^{2} \)
43 \( 1 + (-7.41 + 7.41i)T - 43iT^{2} \)
47 \( 1 + (1.12 + 1.94i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.687 - 2.56i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.12 - 11.6i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.669 - 2.49i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (8.99 - 2.41i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 3.89iT - 71T^{2} \)
73 \( 1 + (-3.46 - 2i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.37 + 2.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.77 - 6.77i)T + 83iT^{2} \)
89 \( 1 + (13.4 - 7.77i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 14.3T + 97T^{2} \)
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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

−11.04298169861475933745119963134, −10.26756736174237769081326172833, −10.00579804511949834857260094717, −8.918396019470392962655582399205, −7.54416533940418072492296111892, −6.78960473335122287449250833651, −5.75039306635202202311034776628, −5.10397446405625068179992182094, −2.79997887684446587809712179538, −1.08725530218116088116906686945, 1.63676478821641159616718060306, 2.47202198930272395329906664733, 4.67649456482840686155742967086, 5.85524905820868047959694785013, 6.63352299252815727804172023412, 8.026389343595825515919801376435, 9.059718792973471052287588051689, 9.766825585310805727729650672157, 10.36603658198424195649687661101, 11.54193879955732669603267527620

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