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Properties

Label 2-336-112.101-c2-0-2
Degree $2$
Conductor $336$
Sign $-0.982 + 0.186i$
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  + (−1.81 + 0.836i)2-s + (−1.67 + 0.448i)3-s + (2.60 − 3.03i)4-s + (−7.08 − 1.89i)5-s + (2.66 − 2.21i)6-s + (3.71 + 5.93i)7-s + (−2.17 + 7.69i)8-s + (2.59 − 1.50i)9-s + (14.4 − 2.47i)10-s + (17.9 − 4.82i)11-s + (−2.98 + 6.25i)12-s + (7.01 + 7.01i)13-s + (−11.7 − 7.66i)14-s + 12.7·15-s + (−2.47 − 15.8i)16-s + (−20.0 − 11.5i)17-s + ⋯
L(s)  = 1  + (−0.908 + 0.418i)2-s + (−0.557 + 0.149i)3-s + (0.650 − 0.759i)4-s + (−1.41 − 0.379i)5-s + (0.444 − 0.369i)6-s + (0.530 + 0.847i)7-s + (−0.272 + 0.962i)8-s + (0.288 − 0.166i)9-s + (1.44 − 0.247i)10-s + (1.63 − 0.438i)11-s + (−0.248 + 0.520i)12-s + (0.539 + 0.539i)13-s + (−0.836 − 0.547i)14-s + 0.847·15-s + (−0.154 − 0.987i)16-s + (−1.17 − 0.680i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.186i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.982 + 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.982 + 0.186i$
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: \((2,\ 336,\ (\ :1),\ -0.982 + 0.186i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0120727 - 0.128372i\)
\(L(\frac12)\) \(\approx\) \(0.0120727 - 0.128372i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.81 - 0.836i)T \)
3 \( 1 + (1.67 - 0.448i)T \)
7 \( 1 + (-3.71 - 5.93i)T \)
good5 \( 1 + (7.08 + 1.89i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (-17.9 + 4.82i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (-7.01 - 7.01i)T + 169iT^{2} \)
17 \( 1 + (20.0 + 11.5i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (0.314 - 1.17i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (29.0 - 16.7i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-3.41 + 3.41i)T - 841iT^{2} \)
31 \( 1 + (15.4 + 8.94i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (14.7 - 54.8i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 64.0T + 1.68e3T^{2} \)
43 \( 1 + (45.2 + 45.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (28.0 - 16.2i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-51.0 + 13.6i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (4.96 + 18.5i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-16.9 + 63.1i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (3.79 + 14.1i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 85.4iT - 5.04e3T^{2} \)
73 \( 1 + (47.0 - 81.4i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (16.3 + 28.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (95.3 + 95.3i)T + 6.88e3iT^{2} \)
89 \( 1 + (-51.5 - 89.3i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 18.6iT - 9.40e3T^{2} \)
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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

−11.65249121368151778075676834316, −11.26324290895156122446965767376, −9.774676660864220674558559707684, −8.731302620213513368382629420700, −8.378076239844424551353465542279, −7.05587997103319349804898441945, −6.25673225611101263108880106054, −4.98007419683014319743777321482, −3.81739371392981491722557584568, −1.57270294994068872039640024409, 0.092773659761406634334098341143, 1.58658770140997285812506969321, 3.71449794826026282608517058283, 4.25803206014449766204385086417, 6.48451528663321205410553658313, 7.09505059608236609536511514526, 8.052048590576930183316002290440, 8.812472958313093180733649459294, 10.25914661054227175234005571212, 10.88543944196450565211326690078

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