↑

Properties

Label 2-338-13.2-c2-0-3
Degree $2$
Conductor $338$
Sign $-0.955 + 0.295i$
Analytic cond. $9.20983$
Root an. cond. $3.03477$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 0.366i)2-s + (2.38 + 4.13i)3-s + (1.73 + i)4-s + (−5.88 + 5.88i)5-s + (−1.74 − 6.52i)6-s + (−0.344 + 0.0922i)7-s + (−1.99 − 2i)8-s + (−6.90 + 11.9i)9-s + (10.2 − 5.88i)10-s + (−0.191 + 0.715i)11-s + 9.55i·12-s + 0.503·14-s + (−38.4 − 10.2i)15-s + (1.99 + 3.46i)16-s + (2.49 + 1.44i)17-s + (13.8 − 13.8i)18-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.795 + 1.37i)3-s + (0.433 + 0.250i)4-s + (−1.17 + 1.17i)5-s + (−0.291 − 1.08i)6-s + (−0.0491 + 0.0131i)7-s + (−0.249 − 0.250i)8-s + (−0.767 + 1.32i)9-s + (1.02 − 0.588i)10-s + (−0.0174 + 0.0650i)11-s + 0.795i·12-s + 0.0359·14-s + (−2.56 − 0.686i)15-s + (0.124 + 0.216i)16-s + (0.146 + 0.0847i)17-s + (0.767 − 0.767i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $-0.955 + 0.295i$
Analytic conductor: \(9.20983\)
Root analytic conductor: \(3.03477\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1),\ -0.955 + 0.295i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.119035 - 0.786510i\)
\(L(\frac12)\) \(\approx\) \(0.119035 - 0.786510i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
13 \( 1 \)
good3 \( 1 + (-2.38 - 4.13i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (5.88 - 5.88i)T - 25iT^{2} \)
7 \( 1 + (0.344 - 0.0922i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (0.191 - 0.715i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (-2.49 - 1.44i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-2.81 - 10.4i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (19.8 - 11.4i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (15.2 + 26.3i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-14.8 + 14.8i)T - 961iT^{2} \)
37 \( 1 + (-10.7 + 40.2i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (-24.8 - 6.66i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (-25.3 - 14.6i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-21.2 - 21.2i)T + 2.20e3iT^{2} \)
53 \( 1 + 85.8T + 2.80e3T^{2} \)
59 \( 1 + (96.6 - 25.9i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (7.73 - 13.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (67.2 + 18.0i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (-25.2 - 94.2i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (-50.9 - 50.9i)T + 5.32e3iT^{2} \)
79 \( 1 - 105.T + 6.24e3T^{2} \)
83 \( 1 + (27.2 - 27.2i)T - 6.88e3iT^{2} \)
89 \( 1 + (22.0 - 82.1i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-5.26 - 19.6i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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

−11.33434777061625184661330885125, −10.81088034895231919066073601277, −9.923045752317467616347744719593, −9.276815859758384278112724543995, −7.983744785393491958257683368740, −7.64861869671563812336049755893, −6.13331275718396816560623807701, −4.28534506484295689355569315584, −3.57271365676808140628089588949, −2.60455849101783566800364872632, 0.42062447433472061680765887751, 1.61701255166901198988410253577, 3.20370523806023858493290398078, 4.76416459670042485592524128204, 6.34421195506594706639738048199, 7.40994157808797179277927193780, 7.995608039790122638241302138782, 8.637898673353842190471018129804, 9.403370750119430238319246919234, 10.95903936173079087352074009903

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