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Properties

Label 2-336-112.53-c1-0-11
Degree $2$
Conductor $336$
Sign $0.753 - 0.657i$
Analytic cond. $2.68297$
Root an. cond. $1.63797$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 0.707i)2-s + (0.258 + 0.965i)3-s + (0.999 − 1.73i)4-s + (−0.990 + 3.69i)5-s + (1 + 0.999i)6-s + (−0.358 + 2.62i)7-s − 2.82i·8-s + (−0.866 + 0.499i)9-s + (1.40 + 5.22i)10-s + (2.33 − 0.624i)11-s + (1.93 + 0.517i)12-s + (−2.41 + 2.41i)13-s + (1.41 + 3.46i)14-s − 3.82·15-s + (−2.00 − 3.46i)16-s + (3.41 − 5.91i)17-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)2-s + (0.149 + 0.557i)3-s + (0.499 − 0.866i)4-s + (−0.443 + 1.65i)5-s + (0.408 + 0.408i)6-s + (−0.135 + 0.990i)7-s − 0.999i·8-s + (−0.288 + 0.166i)9-s + (0.443 + 1.65i)10-s + (0.703 − 0.188i)11-s + (0.557 + 0.149i)12-s + (−0.669 + 0.669i)13-s + (0.377 + 0.925i)14-s − 0.988·15-s + (−0.500 − 0.866i)16-s + (0.828 − 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.753 - 0.657i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.753 - 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93671 + 0.725583i\)
\(L(\frac12)\) \(\approx\) \(1.93671 + 0.725583i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 + (-0.258 - 0.965i)T \)
7 \( 1 + (0.358 - 2.62i)T \)
good5 \( 1 + (0.990 - 3.69i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.33 + 0.624i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (2.41 - 2.41i)T - 13iT^{2} \)
17 \( 1 + (-3.41 + 5.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.73 - 0.732i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.210 - 0.121i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.12 + 6.12i)T - 29iT^{2} \)
31 \( 1 + (0.207 - 0.358i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.732 + 2.73i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 3.41iT - 41T^{2} \)
43 \( 1 + (-7.41 - 7.41i)T + 43iT^{2} \)
47 \( 1 + (1.12 + 1.94i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.56 - 0.687i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (11.6 - 3.12i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.49 - 0.669i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-2.41 - 8.99i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.89iT - 71T^{2} \)
73 \( 1 + (3.46 + 2i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.37 + 2.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.77 + 6.77i)T - 83iT^{2} \)
89 \( 1 + (-13.4 + 7.77i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 14.3T + 97T^{2} \)
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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

−11.81786487022501922771676299167, −10.97027798059607163043195737893, −9.906887865099136587563410326050, −9.348491847223584259157978097466, −7.59435381477321212470058271147, −6.64308239812940838442051723435, −5.69942039742925686914771599414, −4.40255654457860595738017897534, −3.18685174309547378515284171212, −2.55092011449785759405191304355, 1.27474782144383591587783057011, 3.45046116967927594345308739545, 4.45452154607500076662299311177, 5.38438700300467520310216283872, 6.60564401491996044369433650417, 7.72634739353099442571842655918, 8.194732453454146591544547027067, 9.340318107798364885039917715583, 10.72426063596282697188298898066, 12.14559153415619160621057451638

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