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Properties

Label 2-336-7.4-c3-0-17
Degree $2$
Conductor $336$
Sign $0.266 + 0.963i$
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  + (1.5 − 2.59i)3-s + (3 + 5.19i)5-s + (3.5 − 18.1i)7-s + (−4.5 − 7.79i)9-s + (−15 + 25.9i)11-s + 53·13-s + 18·15-s + (42 − 72.7i)17-s + (−48.5 − 84.0i)19-s + (−42 − 36.3i)21-s + (42 + 72.7i)23-s + (44.5 − 77.0i)25-s − 27·27-s − 180·29-s + (89.5 − 155. i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.268 + 0.464i)5-s + (0.188 − 0.981i)7-s + (−0.166 − 0.288i)9-s + (−0.411 + 0.712i)11-s + 1.13·13-s + 0.309·15-s + (0.599 − 1.03i)17-s + (−0.585 − 1.01i)19-s + (−0.436 − 0.377i)21-s + (0.380 + 0.659i)23-s + (0.355 − 0.616i)25-s − 0.192·27-s − 1.15·29-s + (0.518 − 0.898i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.266 + 0.963i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.266 + 0.963i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.109779729\)
\(L(\frac12)\) \(\approx\) \(2.109779729\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.5 + 2.59i)T \)
7 \( 1 + (-3.5 + 18.1i)T \)
good5 \( 1 + (-3 - 5.19i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (15 - 25.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 53T + 2.19e3T^{2} \)
17 \( 1 + (-42 + 72.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (48.5 + 84.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-42 - 72.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 180T + 2.43e4T^{2} \)
31 \( 1 + (-89.5 + 155. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-72.5 - 125. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 126T + 6.89e4T^{2} \)
43 \( 1 - 325T + 7.95e4T^{2} \)
47 \( 1 + (183 + 316. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-384 + 665. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (132 - 228. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (409 + 708. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (261.5 - 452. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 342T + 3.57e5T^{2} \)
73 \( 1 + (-21.5 + 37.2i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (585.5 + 1.01e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 810T + 5.71e5T^{2} \)
89 \( 1 + (-300 - 519. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 386T + 9.12e5T^{2} \)
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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

−10.95378288510854037092342444626, −10.05225968005947847295314871523, −9.091037685216651748442886691267, −7.86525666662873790087089655565, −7.17687358989386597530319053889, −6.28091350933002804561453977160, −4.88001513111404739788044642665, −3.58828673497421251871083209256, −2.28726387124614889666923272081, −0.78304345843527110081119918259, 1.47887785997626232981971450832, 2.98704771627784965502596770183, 4.17774080601078504929663272650, 5.57581922187087039243290952751, 6.04577420220267635039147714873, 7.88321947704415834794264872501, 8.651635375981773251661134814541, 9.211961255531357933229388401213, 10.51951037909503159526144625777, 11.07716308288106679009702640315

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