Properties

Label 2-338-13.6-c2-0-21
Degree $2$
Conductor $338$
Sign $0.444 + 0.895i$
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  + (0.366 + 1.36i)2-s + (2.76 − 4.78i)3-s + (−1.73 + i)4-s + (3.04 − 3.04i)5-s + (7.54 + 2.02i)6-s + (−0.959 + 3.58i)7-s + (−2 − 1.99i)8-s + (−10.7 − 18.6i)9-s + (5.26 + 3.04i)10-s + (14.8 − 3.96i)11-s + 11.0i·12-s − 5.24·14-s + (−6.14 − 22.9i)15-s + (1.99 − 3.46i)16-s + (−2.62 + 1.51i)17-s + (21.5 − 21.5i)18-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (0.920 − 1.59i)3-s + (−0.433 + 0.250i)4-s + (0.608 − 0.608i)5-s + (1.25 + 0.336i)6-s + (−0.137 + 0.511i)7-s + (−0.250 − 0.249i)8-s + (−1.19 − 2.07i)9-s + (0.526 + 0.304i)10-s + (1.34 − 0.360i)11-s + 0.920i·12-s − 0.374·14-s + (−0.409 − 1.52i)15-s + (0.124 − 0.216i)16-s + (−0.154 + 0.0889i)17-s + (1.19 − 1.19i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $0.444 + 0.895i$
Analytic conductor: \(9.20983\)
Root analytic conductor: \(3.03477\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1),\ 0.444 + 0.895i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.16442 - 1.34232i\)
\(L(\frac12)\) \(\approx\) \(2.16442 - 1.34232i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
13 \( 1 \)
good3 \( 1 + (-2.76 + 4.78i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (-3.04 + 3.04i)T - 25iT^{2} \)
7 \( 1 + (0.959 - 3.58i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (-14.8 + 3.96i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (2.62 - 1.51i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-7.43 - 1.99i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (28.8 + 16.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-20.3 + 35.1i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (15.1 - 15.1i)T - 961iT^{2} \)
37 \( 1 + (10.2 - 2.73i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (-0.256 - 0.957i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (-19.8 + 11.4i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (9.74 + 9.74i)T + 2.20e3iT^{2} \)
53 \( 1 - 6.78T + 2.80e3T^{2} \)
59 \( 1 + (18.4 - 68.7i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-35.9 - 62.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-34.4 - 128. i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (-75.8 - 20.3i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (-49.0 - 49.0i)T + 5.32e3iT^{2} \)
79 \( 1 - 7.91T + 6.24e3T^{2} \)
83 \( 1 + (-59.1 + 59.1i)T - 6.88e3iT^{2} \)
89 \( 1 + (-21.0 + 5.65i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (74.9 + 20.0i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−11.76031574975976654678052035692, −9.716751294603199545017656375471, −8.837719396587058180692272446002, −8.410458405683293504480903394432, −7.27799989216758614704104208492, −6.37790788690406697217301854592, −5.72313786588082503269720299269, −3.89756589303794155087376544706, −2.40461714286481379167220257863, −1.11262544316113491675690541003, 2.07352523251135909324896123299, 3.38399566954352458795844113774, 4.02639797352937801108492002366, 5.13261403228674449114302321172, 6.54032079035133820793006993089, 8.050018387908059648579736722289, 9.316917106946196484766977316761, 9.585618073091943727038322644134, 10.46104238796636025679674144083, 11.10267126004631414935710300731

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