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Properties

Label 2-336-84.11-c3-0-8
Degree $2$
Conductor $336$
Sign $0.129 - 0.991i$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.34 − 4.63i)3-s + (8.90 − 5.14i)5-s + (9.06 + 16.1i)7-s + (−15.9 + 21.7i)9-s + (−23.0 + 40.0i)11-s + 26.5·13-s + (−44.7 − 29.2i)15-s + (−102. − 59.4i)17-s + (−142. + 82.5i)19-s + (53.5 − 79.9i)21-s + (65.7 + 113. i)23-s + (−9.59 + 16.6i)25-s + (138. + 22.8i)27-s + 15.1i·29-s + (−101. − 58.4i)31-s + ⋯
L(s)  = 1  + (−0.452 − 0.891i)3-s + (0.796 − 0.460i)5-s + (0.489 + 0.872i)7-s + (−0.591 + 0.806i)9-s + (−0.633 + 1.09i)11-s + 0.566·13-s + (−0.770 − 0.502i)15-s + (−1.46 − 0.848i)17-s + (−1.72 + 0.996i)19-s + (0.556 − 0.830i)21-s + (0.596 + 1.03i)23-s + (−0.0767 + 0.132i)25-s + (0.986 + 0.162i)27-s + 0.0968i·29-s + (−0.586 − 0.338i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.129 - 0.991i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.129 - 0.991i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.011918940\)
\(L(\frac12)\) \(\approx\) \(1.011918940\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (2.34 + 4.63i)T \)
7 \( 1 + (-9.06 - 16.1i)T \)
good5 \( 1 + (-8.90 + 5.14i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (23.0 - 40.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 26.5T + 2.19e3T^{2} \)
17 \( 1 + (102. + 59.4i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (142. - 82.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-65.7 - 113. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 15.1iT - 2.43e4T^{2} \)
31 \( 1 + (101. + 58.4i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-113. - 196. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 220. iT - 6.89e4T^{2} \)
43 \( 1 - 338. iT - 7.95e4T^{2} \)
47 \( 1 + (169. + 293. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-660. - 381. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (238. - 412. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (148. + 257. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (596. + 344. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 700.T + 3.57e5T^{2} \)
73 \( 1 + (155. - 268. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (248. - 143. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 906.T + 5.71e5T^{2} \)
89 \( 1 + (-616. + 355. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 354.T + 9.12e5T^{2} \)
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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

−11.43730410004864222434517740358, −10.57768814947705529009867070846, −9.318439700229927486077227514624, −8.535199093739595068527862025581, −7.49091693142598302353200932499, −6.37971755396125311693470454983, −5.52870181806095799670232509720, −4.64444965800023584325818588374, −2.35021634631816599402886962062, −1.66021395865460227868005710817, 0.35842166016936132136713465064, 2.38441297331688946225842765244, 3.88345335159383848793334313414, 4.80902372404003739687394847031, 6.09243953691833300320720166236, 6.67575206893022583130211185199, 8.390885514680262795515354488076, 9.007491821000158930761249810970, 10.52232259128952496172038421367, 10.66700943645799211332983356640

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