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Properties

Label 2-338-13.11-c2-0-4
Degree $2$
Conductor $338$
Sign $0.283 - 0.958i$
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 + (1.92 + 3.33i)3-s + (−1.73 − i)4-s + (3.77 + 3.77i)5-s + (5.25 − 1.40i)6-s + (2.65 + 9.91i)7-s + (−2 + 1.99i)8-s + (−2.90 + 5.02i)9-s + (6.53 − 3.77i)10-s + (−10.1 − 2.71i)11-s − 7.69i·12-s + 14.5·14-s + (−5.31 + 19.8i)15-s + (1.99 + 3.46i)16-s + (−4.23 − 2.44i)17-s + (5.80 + 5.80i)18-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.641 + 1.11i)3-s + (−0.433 − 0.250i)4-s + (0.754 + 0.754i)5-s + (0.875 − 0.234i)6-s + (0.379 + 1.41i)7-s + (−0.250 + 0.249i)8-s + (−0.322 + 0.558i)9-s + (0.653 − 0.377i)10-s + (−0.919 − 0.246i)11-s − 0.641i·12-s + 1.03·14-s + (−0.354 + 1.32i)15-s + (0.124 + 0.216i)16-s + (−0.249 − 0.143i)17-s + (0.322 + 0.322i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 - 0.958i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.283 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $0.283 - 0.958i$
Analytic conductor: \(9.20983\)
Root analytic conductor: \(3.03477\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1),\ 0.283 - 0.958i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.81879 + 1.35871i\)
\(L(\frac12)\) \(\approx\) \(1.81879 + 1.35871i\)
\(L(2)\) 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 \( 1 + (-1.92 - 3.33i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-3.77 - 3.77i)T + 25iT^{2} \)
7 \( 1 + (-2.65 - 9.91i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (10.1 + 2.71i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (4.23 + 2.44i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (25.5 - 6.83i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (-17.2 + 9.97i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (7.15 + 12.3i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-19.0 - 19.0i)T + 961iT^{2} \)
37 \( 1 + (-58.6 - 15.7i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-1.29 + 4.83i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-10.3 - 5.98i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (7.59 - 7.59i)T - 2.20e3iT^{2} \)
53 \( 1 - 77.0T + 2.80e3T^{2} \)
59 \( 1 + (-16.2 - 60.7i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-28.1 + 48.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (1.58 - 5.90i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-55.2 + 14.7i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-12.7 + 12.7i)T - 5.32e3iT^{2} \)
79 \( 1 - 7.98T + 6.24e3T^{2} \)
83 \( 1 + (35.8 + 35.8i)T + 6.88e3iT^{2} \)
89 \( 1 + (78.2 + 20.9i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-52.9 + 14.1i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−11.23641800551514056376595272401, −10.49327877824339952417356344708, −9.830809909566671668886709111866, −8.911708206835339331537380893733, −8.266675444315516964452241862122, −6.36757224426237677382501530895, −5.37462775237180495221258569136, −4.33347318865074687002553407879, −2.83833346126425108080156264030, −2.36531714043997040363532203168, 0.956296877053852724624020629193, 2.33973325832591182823160237062, 4.19624629675247973889155188021, 5.22996979796725080068600983116, 6.54254375862083812682180916301, 7.38007175621655445074408471826, 8.058604020868430671699672508006, 8.932004785216744884151599514944, 10.07899447780193842014288652298, 11.10065547375191290368852565903

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