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Properties

Label 2-336-112.109-c1-0-15
Degree $2$
Conductor $336$
Sign $0.0829 + 0.996i$
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  + (−1.37 − 0.338i)2-s + (−0.965 + 0.258i)3-s + (1.77 + 0.929i)4-s + (0.552 + 0.147i)5-s + (1.41 − 0.0283i)6-s + (−2.64 − 0.171i)7-s + (−2.11 − 1.87i)8-s + (0.866 − 0.499i)9-s + (−0.708 − 0.390i)10-s + (−1.26 − 4.70i)11-s + (−1.95 − 0.439i)12-s + (3.13 + 3.13i)13-s + (3.56 + 1.12i)14-s − 0.571·15-s + (2.27 + 3.29i)16-s + (1.06 − 1.83i)17-s + ⋯
L(s)  = 1  + (−0.970 − 0.239i)2-s + (−0.557 + 0.149i)3-s + (0.885 + 0.464i)4-s + (0.246 + 0.0661i)5-s + (0.577 − 0.0115i)6-s + (−0.997 − 0.0647i)7-s + (−0.748 − 0.663i)8-s + (0.288 − 0.166i)9-s + (−0.223 − 0.123i)10-s + (−0.380 − 1.41i)11-s + (−0.563 − 0.126i)12-s + (0.870 + 0.870i)13-s + (0.953 + 0.301i)14-s − 0.147·15-s + (0.567 + 0.823i)16-s + (0.257 − 0.445i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0829 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0829 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.0829 + 0.996i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.0829 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.410526 - 0.377793i\)
\(L(\frac12)\) \(\approx\) \(0.410526 - 0.377793i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 + 0.338i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (2.64 + 0.171i)T \)
good5 \( 1 + (-0.552 - 0.147i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.26 + 4.70i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-3.13 - 3.13i)T + 13iT^{2} \)
17 \( 1 + (-1.06 + 1.83i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.705 + 2.63i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.94 + 2.27i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.07 + 6.07i)T + 29iT^{2} \)
31 \( 1 + (-5.31 + 9.21i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.78 - 1.81i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 6.70iT - 41T^{2} \)
43 \( 1 + (4.07 - 4.07i)T - 43iT^{2} \)
47 \( 1 + (3.63 + 6.29i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.10 - 7.84i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.33 - 8.73i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.16 + 8.06i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (10.8 - 2.90i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 6.12iT - 71T^{2} \)
73 \( 1 + (6.80 + 3.93i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.530 + 0.919i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.20 - 4.20i)T + 83iT^{2} \)
89 \( 1 + (-3.57 + 2.06i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.40T + 97T^{2} \)
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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

−11.35130465172001831095005781429, −10.34759208431164750373028771501, −9.529517421705783839430346972064, −8.775960268808071765186660192391, −7.60017127298996088852851781667, −6.38640314776320833940813298364, −5.91029632978111536597310655084, −3.91770082151807038359801845767, −2.66585645114402308904322425103, −0.59425557124243322727024298905, 1.51644031582157758318990491827, 3.24365343068217259285803779541, 5.23052616250166632220653891584, 6.11939187166220107396361352180, 7.04269342403099507933159614904, 7.910324741572622931519025575913, 9.133793908803251555798375380845, 9.999520613092840044338823746659, 10.51341298248381690349198724960, 11.61847326981555123222231214241

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