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Properties

Label 2-336-84.23-c3-0-41
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: $\chi_{336} (191, \cdot )$
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

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

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