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Properties

Label 2-338-13.7-c2-0-22
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

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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: $\chi_{338} (319, \cdot )$
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 \( 1 + (-5.26 + 19.6i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

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

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