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Properties

Label 2-336-112.37-c1-0-26
Degree $2$
Conductor $336$
Sign $-0.999 + 0.0428i$
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  + (−1.30 − 0.537i)2-s + (−0.965 − 0.258i)3-s + (1.42 + 1.40i)4-s + (−1.52 + 0.408i)5-s + (1.12 + 0.857i)6-s + (1.09 − 2.40i)7-s + (−1.10 − 2.60i)8-s + (0.866 + 0.499i)9-s + (2.21 + 0.284i)10-s + (−0.686 + 2.56i)11-s + (−1.01 − 1.72i)12-s + (−0.948 + 0.948i)13-s + (−2.72 + 2.56i)14-s + 1.57·15-s + (0.0490 + 3.99i)16-s + (−2.32 − 4.02i)17-s + ⋯
L(s)  = 1  + (−0.925 − 0.379i)2-s + (−0.557 − 0.149i)3-s + (0.711 + 0.702i)4-s + (−0.681 + 0.182i)5-s + (0.459 + 0.350i)6-s + (0.412 − 0.910i)7-s + (−0.391 − 0.920i)8-s + (0.288 + 0.166i)9-s + (0.699 + 0.0899i)10-s + (−0.207 + 0.772i)11-s + (−0.291 − 0.498i)12-s + (−0.262 + 0.262i)13-s + (−0.727 + 0.685i)14-s + 0.407·15-s + (0.0122 + 0.999i)16-s + (−0.563 − 0.975i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0428i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.999 + 0.0428i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.999 + 0.0428i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00406857 - 0.189708i\)
\(L(\frac12)\) \(\approx\) \(0.00406857 - 0.189708i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 + 0.537i)T \)
3 \( 1 + (0.965 + 0.258i)T \)
7 \( 1 + (-1.09 + 2.40i)T \)
good5 \( 1 + (1.52 - 0.408i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.686 - 2.56i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.948 - 0.948i)T - 13iT^{2} \)
17 \( 1 + (2.32 + 4.02i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.53 + 5.73i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (5.02 + 2.90i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.50 - 6.50i)T - 29iT^{2} \)
31 \( 1 + (1.32 + 2.29i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (10.3 - 2.76i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 3.43iT - 41T^{2} \)
43 \( 1 + (-3.31 - 3.31i)T + 43iT^{2} \)
47 \( 1 + (-3.52 + 6.10i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.81 + 10.5i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.68 + 6.29i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.738 + 2.75i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-12.7 - 3.42i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 2.52iT - 71T^{2} \)
73 \( 1 + (7.51 - 4.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.62 + 2.80i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.40 - 2.40i)T - 83iT^{2} \)
89 \( 1 + (-9.99 - 5.76i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.78T + 97T^{2} \)
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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

−11.15628838202577229068320543767, −10.30515972278732427725921923333, −9.375574499785927108496255018728, −8.194834204005269667062218926329, −7.16760849704658379597869957428, −6.88893580129530191885780234583, −4.92009033576418816370636470244, −3.80314371305038017330355747482, −2.06711585304097888046450451459, −0.18161716768345310374067154872, 1.96460984091621975788501150283, 3.94458065210107348188542466724, 5.60675950424428807322782915224, 6.00259349878785530520039262035, 7.55516845583550015686134103069, 8.241483387044301080856669770772, 9.047633094484578117749758510100, 10.23509435415757086135068270444, 10.96528986886746726455691792218, 11.86542601108588151058151580726

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