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Properties

Label 2-336-112.101-c2-0-16
Degree $2$
Conductor $336$
Sign $0.368 + 0.929i$
Analytic cond. $9.15533$
Root an. cond. $3.02577$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.77 − 0.930i)2-s + (−1.67 + 0.448i)3-s + (2.26 + 3.29i)4-s + (−6.15 − 1.64i)5-s + (3.37 + 0.763i)6-s + (−2.07 + 6.68i)7-s + (−0.949 − 7.94i)8-s + (2.59 − 1.50i)9-s + (9.36 + 8.64i)10-s + (−13.9 + 3.73i)11-s + (−5.27 − 4.49i)12-s + (3.91 + 3.91i)13-s + (9.89 − 9.90i)14-s + 11.0·15-s + (−5.71 + 14.9i)16-s + (4.41 + 2.54i)17-s + ⋯
L(s)  = 1  + (−0.885 − 0.465i)2-s + (−0.557 + 0.149i)3-s + (0.567 + 0.823i)4-s + (−1.23 − 0.329i)5-s + (0.563 + 0.127i)6-s + (−0.296 + 0.955i)7-s + (−0.118 − 0.992i)8-s + (0.288 − 0.166i)9-s + (0.936 + 0.864i)10-s + (−1.26 + 0.339i)11-s + (−0.439 − 0.374i)12-s + (0.301 + 0.301i)13-s + (0.706 − 0.707i)14-s + 0.735·15-s + (−0.356 + 0.934i)16-s + (0.259 + 0.149i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.368 + 0.929i$
Analytic conductor: \(9.15533\)
Root analytic conductor: \(3.02577\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1),\ 0.368 + 0.929i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.331848 - 0.225437i\)
\(L(\frac12)\) \(\approx\) \(0.331848 - 0.225437i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.77 + 0.930i)T \)
3 \( 1 + (1.67 - 0.448i)T \)
7 \( 1 + (2.07 - 6.68i)T \)
good5 \( 1 + (6.15 + 1.64i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (13.9 - 3.73i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (-3.91 - 3.91i)T + 169iT^{2} \)
17 \( 1 + (-4.41 - 2.54i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-3.46 + 12.9i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (9.93 - 5.73i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-24.4 + 24.4i)T - 841iT^{2} \)
31 \( 1 + (-25.9 - 14.9i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-11.1 + 41.7i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 15.6T + 1.68e3T^{2} \)
43 \( 1 + (37.5 + 37.5i)T + 1.84e3iT^{2} \)
47 \( 1 + (-49.4 + 28.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (25.9 - 6.94i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-3.76 - 14.0i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-26.7 + 99.7i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (-21.6 - 80.8i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 127. iT - 5.04e3T^{2} \)
73 \( 1 + (-19.0 + 32.9i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-61.8 - 107. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-51.1 - 51.1i)T + 6.88e3iT^{2} \)
89 \( 1 + (-4.04 - 7.00i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 25.8iT - 9.40e3T^{2} \)
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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

−11.18274950253916329855886803122, −10.31270334308064744015081646025, −9.327489652793808128993456363442, −8.333807032828315755473051767275, −7.67442214376223896612452891413, −6.48835820611579560638436311090, −5.10201879407735046993974733538, −3.81619086554758978887074606255, −2.45678157151692382593695294741, −0.39465891086243172275944486008, 0.831033283873884416972771574589, 3.09524336180238477833447408466, 4.61282341561285652579394308328, 5.92458831597888139659087906418, 6.94814138365727527071289387816, 7.81181101097596295703010575351, 8.242596806953349587501038226460, 9.932397609284971234933809202312, 10.53060527916848634631222813198, 11.23945470981459791867098007547

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