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Properties

Label 2-336-112.101-c2-0-29
Degree $2$
Conductor $336$
Sign $0.123 - 0.992i$
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  + (1.50 + 1.31i)2-s + (−1.67 + 0.448i)3-s + (0.556 + 3.96i)4-s + (7.94 + 2.12i)5-s + (−3.11 − 1.51i)6-s + (−0.869 − 6.94i)7-s + (−4.35 + 6.70i)8-s + (2.59 − 1.50i)9-s + (9.19 + 13.6i)10-s + (12.7 − 3.40i)11-s + (−2.70 − 6.37i)12-s + (17.3 + 17.3i)13-s + (7.80 − 11.6i)14-s − 14.2·15-s + (−15.3 + 4.41i)16-s + (−19.2 − 11.1i)17-s + ⋯
L(s)  = 1  + (0.754 + 0.656i)2-s + (−0.557 + 0.149i)3-s + (0.139 + 0.990i)4-s + (1.58 + 0.425i)5-s + (−0.518 − 0.253i)6-s + (−0.124 − 0.992i)7-s + (−0.544 + 0.838i)8-s + (0.288 − 0.166i)9-s + (0.919 + 1.36i)10-s + (1.15 − 0.309i)11-s + (−0.225 − 0.531i)12-s + (1.33 + 1.33i)13-s + (0.557 − 0.830i)14-s − 0.949·15-s + (−0.961 + 0.275i)16-s + (−1.13 − 0.654i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 - 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.123 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.123 - 0.992i$
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: \((2,\ 336,\ (\ :1),\ 0.123 - 0.992i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.08986 + 1.84587i\)
\(L(\frac12)\) \(\approx\) \(2.08986 + 1.84587i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.50 - 1.31i)T \)
3 \( 1 + (1.67 - 0.448i)T \)
7 \( 1 + (0.869 + 6.94i)T \)
good5 \( 1 + (-7.94 - 2.12i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (-12.7 + 3.40i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (-17.3 - 17.3i)T + 169iT^{2} \)
17 \( 1 + (19.2 + 11.1i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (0.125 - 0.467i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (2.60 - 1.50i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (16.6 - 16.6i)T - 841iT^{2} \)
31 \( 1 + (38.4 + 22.1i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (5.78 - 21.6i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 18.3T + 1.68e3T^{2} \)
43 \( 1 + (-40.8 - 40.8i)T + 1.84e3iT^{2} \)
47 \( 1 + (-19.1 + 11.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (10.7 - 2.86i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (11.6 + 43.4i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-13.1 + 48.9i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (20.3 + 75.7i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 69.9iT - 5.04e3T^{2} \)
73 \( 1 + (-59.2 + 102. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (33.6 + 58.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (2.88 + 2.88i)T + 6.88e3iT^{2} \)
89 \( 1 + (22.2 + 38.4i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 128. iT - 9.40e3T^{2} \)
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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

−11.26418815662118053585003239566, −11.04714597056668571677605144903, −9.515458337322929522755297153150, −8.955320254151783051949516106002, −7.15497615302875706167568468530, −6.45103946360731746973574579169, −5.98613219992623936788819671191, −4.58816650849243177561948437504, −3.61459311927261434029927507402, −1.76355450684506155632143266665, 1.31402046530373932733063791224, 2.32413694028062746782165053093, 3.97960519989198780404492363198, 5.49309640990050587919872982654, 5.82330636508075054507662783098, 6.63829729532998284563884600952, 8.800531823412705548092367077354, 9.345011382074461421249195167192, 10.45045458299559968081406652188, 11.11077488361935036717758528141

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