Properties

Label 2-336-112.101-c2-0-31
Degree $2$
Conductor $336$
Sign $0.117 + 0.993i$
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  + (0.480 − 1.94i)2-s + (−1.67 + 0.448i)3-s + (−3.53 − 1.86i)4-s + (7.99 + 2.14i)5-s + (0.0670 + 3.46i)6-s + (−6.99 + 0.309i)7-s + (−5.31 + 5.97i)8-s + (2.59 − 1.50i)9-s + (8.00 − 14.4i)10-s + (16.5 − 4.43i)11-s + (6.75 + 1.53i)12-s + (−6.50 − 6.50i)13-s + (−2.75 + 13.7i)14-s − 14.3·15-s + (9.04 + 13.1i)16-s + (20.4 + 11.8i)17-s + ⋯
L(s)  = 1  + (0.240 − 0.970i)2-s + (−0.557 + 0.149i)3-s + (−0.884 − 0.466i)4-s + (1.59 + 0.428i)5-s + (0.0111 + 0.577i)6-s + (−0.999 + 0.0441i)7-s + (−0.664 + 0.746i)8-s + (0.288 − 0.166i)9-s + (0.800 − 1.44i)10-s + (1.50 − 0.403i)11-s + (0.563 + 0.127i)12-s + (−0.500 − 0.500i)13-s + (−0.196 + 0.980i)14-s − 0.956·15-s + (0.565 + 0.824i)16-s + (1.20 + 0.695i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.117 + 0.993i$
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: \((2,\ 336,\ (\ :1),\ 0.117 + 0.993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.33224 - 1.18432i\)
\(L(\frac12)\) \(\approx\) \(1.33224 - 1.18432i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.480 + 1.94i)T \)
3 \( 1 + (1.67 - 0.448i)T \)
7 \( 1 + (6.99 - 0.309i)T \)
good5 \( 1 + (-7.99 - 2.14i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (-16.5 + 4.43i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (6.50 + 6.50i)T + 169iT^{2} \)
17 \( 1 + (-20.4 - 11.8i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-7.48 + 27.9i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (-15.3 + 8.87i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-15.4 + 15.4i)T - 841iT^{2} \)
31 \( 1 + (39.3 + 22.7i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (1.01 - 3.77i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 0.282T + 1.68e3T^{2} \)
43 \( 1 + (18.0 + 18.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (18.8 - 10.8i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-61.2 + 16.4i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (5.56 + 20.7i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (8.61 - 32.1i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (-18.5 - 69.0i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 26.8iT - 5.04e3T^{2} \)
73 \( 1 + (24.8 - 43.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-47.2 - 81.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-89.8 - 89.8i)T + 6.88e3iT^{2} \)
89 \( 1 + (52.0 + 90.0i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 37.2iT - 9.40e3T^{2} \)
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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

−11.04170698164084846222321928361, −10.15009679688755011917707321522, −9.613584354166674570510045596116, −8.943619423708043296664410504508, −6.82987565553705377926557490458, −6.01162059690217796284039056016, −5.26027416379937537640450088260, −3.67801719626208521566125896062, −2.56394486956840627583068519401, −1.00543522001357433266494227728, 1.36319266242156106458525744905, 3.49770366884194914403617872868, 4.99926985173556045401981034151, 5.80180911034710877355980527089, 6.54957374242585809390134832373, 7.31990438565160994821411737073, 9.017309448283793734284815032755, 9.546112446723158850600108783173, 10.15472370691804334045772323418, 12.08628377506119712564245378443

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