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Properties

Label 2-336-48.35-c1-0-32
Degree $2$
Conductor $336$
Sign $0.677 + 0.735i$
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  + (−0.809 + 1.15i)2-s + (0.594 − 1.62i)3-s + (−0.687 − 1.87i)4-s + (0.0563 + 0.0563i)5-s + (1.40 + 2.00i)6-s + 7-s + (2.73 + 0.723i)8-s + (−2.29 − 1.93i)9-s + (−0.110 + 0.0196i)10-s + (0.982 − 0.982i)11-s + (−3.46 + 0.00187i)12-s + (−0.649 − 0.649i)13-s + (−0.809 + 1.15i)14-s + (0.125 − 0.0581i)15-s + (−3.05 + 2.58i)16-s − 4.55i·17-s + ⋯
L(s)  = 1  + (−0.572 + 0.819i)2-s + (0.343 − 0.939i)3-s + (−0.343 − 0.939i)4-s + (0.0251 + 0.0251i)5-s + (0.573 + 0.819i)6-s + 0.377·7-s + (0.966 + 0.255i)8-s + (−0.764 − 0.645i)9-s + (−0.0350 + 0.00622i)10-s + (0.296 − 0.296i)11-s + (−0.999 + 0.000541i)12-s + (−0.180 − 0.180i)13-s + (−0.216 + 0.309i)14-s + (0.0323 − 0.0150i)15-s + (−0.763 + 0.645i)16-s − 1.10i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.677 + 0.735i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.677 + 0.735i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.967225 - 0.424117i\)
\(L(\frac12)\) \(\approx\) \(0.967225 - 0.424117i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 1.15i)T \)
3 \( 1 + (-0.594 + 1.62i)T \)
7 \( 1 - T \)
good5 \( 1 + (-0.0563 - 0.0563i)T + 5iT^{2} \)
11 \( 1 + (-0.982 + 0.982i)T - 11iT^{2} \)
13 \( 1 + (0.649 + 0.649i)T + 13iT^{2} \)
17 \( 1 + 4.55iT - 17T^{2} \)
19 \( 1 + (-2.11 + 2.11i)T - 19iT^{2} \)
23 \( 1 + 2.24iT - 23T^{2} \)
29 \( 1 + (-4.36 + 4.36i)T - 29iT^{2} \)
31 \( 1 - 4.82iT - 31T^{2} \)
37 \( 1 + (1.58 - 1.58i)T - 37iT^{2} \)
41 \( 1 + 5.28T + 41T^{2} \)
43 \( 1 + (-7.65 - 7.65i)T + 43iT^{2} \)
47 \( 1 - 1.18T + 47T^{2} \)
53 \( 1 + (-9.36 - 9.36i)T + 53iT^{2} \)
59 \( 1 + (3.21 - 3.21i)T - 59iT^{2} \)
61 \( 1 + (-0.0261 - 0.0261i)T + 61iT^{2} \)
67 \( 1 + (7.39 - 7.39i)T - 67iT^{2} \)
71 \( 1 + 2.08iT - 71T^{2} \)
73 \( 1 + 14.0iT - 73T^{2} \)
79 \( 1 - 14.7iT - 79T^{2} \)
83 \( 1 + (2.06 + 2.06i)T + 83iT^{2} \)
89 \( 1 + 6.33T + 89T^{2} \)
97 \( 1 + 7.26T + 97T^{2} \)
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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

−11.47548589279397196968682254958, −10.32960419099914604299415565773, −9.218634291874332527891274384814, −8.485432360375079765493009396377, −7.60628140422850716216952806062, −6.81833033049984358444247086334, −5.88344624588765216384629894724, −4.65223597864014961659929033542, −2.64550217156123530683462025665, −0.933992745115138111584984517298, 1.87213388286043050544835693220, 3.37467772292351822643015043512, 4.26510145163932212934592538637, 5.46736531128513366511813532211, 7.26325670576805374463070737485, 8.291391059167340615282199936749, 9.040273504727456148420603547053, 9.903554437263917680000079440341, 10.61489471739997769063664110506, 11.45556722790178909838406306390

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