Properties

Label 2-418-19.6-c1-0-1
Degree $2$
Conductor $418$
Sign $-0.935 - 0.352i$
Analytic cond. $3.33774$
Root an. cond. $1.82695$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.361 + 0.131i)3-s + (0.173 − 0.984i)4-s + (−0.0315 − 0.179i)5-s + (−0.361 + 0.131i)6-s + (−2.25 + 3.89i)7-s + (0.500 + 0.866i)8-s + (−2.18 − 1.83i)9-s + (0.139 + 0.116i)10-s + (0.5 + 0.866i)11-s + (0.192 − 0.333i)12-s + (0.715 − 0.260i)13-s + (−0.781 − 4.43i)14-s + (0.0121 − 0.0689i)15-s + (−0.939 − 0.342i)16-s + (−4.94 + 4.15i)17-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (0.208 + 0.0760i)3-s + (0.0868 − 0.492i)4-s + (−0.0141 − 0.0801i)5-s + (−0.147 + 0.0537i)6-s + (−0.850 + 1.47i)7-s + (0.176 + 0.306i)8-s + (−0.728 − 0.611i)9-s + (0.0440 + 0.0369i)10-s + (0.150 + 0.261i)11-s + (0.0555 − 0.0962i)12-s + (0.198 − 0.0722i)13-s + (−0.208 − 1.18i)14-s + (0.00314 − 0.0178i)15-s + (−0.234 − 0.0855i)16-s + (−1.20 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.935 - 0.352i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.935 - 0.352i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0960268 + 0.527096i\)
\(L(\frac12)\) \(\approx\) \(0.0960268 + 0.527096i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 - 0.642i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (4.35 - 0.0166i)T \)
good3 \( 1 + (-0.361 - 0.131i)T + (2.29 + 1.92i)T^{2} \)
5 \( 1 + (0.0315 + 0.179i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (2.25 - 3.89i)T + (-3.5 - 6.06i)T^{2} \)
13 \( 1 + (-0.715 + 0.260i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (4.94 - 4.15i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (0.621 - 3.52i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-1.98 - 1.66i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (3.72 - 6.45i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.57T + 37T^{2} \)
41 \( 1 + (9.71 + 3.53i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.56 - 8.89i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (5.75 + 4.83i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-0.256 + 1.45i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-3.39 + 2.84i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.172 - 0.980i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-9.17 - 7.69i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.29 + 7.31i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-5.67 - 2.06i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-11.5 - 4.20i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-6.70 + 11.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.47 - 0.899i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (2.58 - 2.16i)T + (16.8 - 95.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.57299720505922677285313190664, −10.54735562610427612527810157154, −9.477327020310380560859749562304, −8.749591410365211933376104945650, −8.402564121229072599863470848184, −6.64357609139720941569436041807, −6.24997468342142639989097415295, −5.10759386921456607237646562630, −3.45823206468522665655384511023, −2.16560295090841470086115459211, 0.37393728065939573320899366741, 2.38083091553492541284016180829, 3.56005317142665897317366396003, 4.66106106376386022943785349048, 6.41269791495597243655311690677, 7.11217502552817089786901652396, 8.185324592339537292811000103473, 9.028364541433407875671020964256, 9.979023923292524232679456452707, 10.88160108369978660576434456858

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