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Properties

Label 2-338-13.11-c2-0-21
Degree $2$
Conductor $338$
Sign $-0.875 - 0.483i$
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 + (−1.73 − i)4-s + (−3 − 3i)5-s + (0.732 + 2.73i)7-s + (−2 + 1.99i)8-s + (4.5 − 7.79i)9-s + (−5.19 + 3i)10-s + (−8.19 − 2.19i)11-s + 4·14-s + (1.99 + 3.46i)16-s + (−5.19 − 3i)17-s + (−9 − 9i)18-s + (−35.5 + 9.51i)19-s + (2.19 + 8.19i)20-s + (−6 + 10.3i)22-s + (−20.7 + 12i)23-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.433 − 0.250i)4-s + (−0.600 − 0.600i)5-s + (0.104 + 0.390i)7-s + (−0.250 + 0.249i)8-s + (0.5 − 0.866i)9-s + (−0.519 + 0.300i)10-s + (−0.745 − 0.199i)11-s + 0.285·14-s + (0.124 + 0.216i)16-s + (−0.305 − 0.176i)17-s + (−0.5 − 0.5i)18-s + (−1.86 + 0.500i)19-s + (0.109 + 0.409i)20-s + (−0.272 + 0.472i)22-s + (−0.903 + 0.521i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 - 0.483i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $-0.875 - 0.483i$
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: \((2,\ 338,\ (\ :1),\ -0.875 - 0.483i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.132555 + 0.514597i\)
\(L(\frac12)\) \(\approx\) \(0.132555 + 0.514597i\)
\(L(2)\) 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 \( 1 + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (3 + 3i)T + 25iT^{2} \)
7 \( 1 + (-0.732 - 2.73i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (8.19 + 2.19i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (5.19 + 3i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (35.5 - 9.51i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (20.7 - 12i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-24 - 41.5i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (14 + 14i)T + 961iT^{2} \)
37 \( 1 + (50.5 + 13.5i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (3.29 - 12.2i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (31.1 + 18i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-42 + 42i)T - 2.20e3iT^{2} \)
53 \( 1 - 30T + 2.80e3T^{2} \)
59 \( 1 + (19.7 + 73.7i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-9 + 15.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (8.05 - 30.0i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (8.19 - 2.19i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-17 + 17i)T - 5.32e3iT^{2} \)
79 \( 1 + 108T + 6.24e3T^{2} \)
83 \( 1 + (-78 - 78i)T + 6.88e3iT^{2} \)
89 \( 1 + (-12.2 - 3.29i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-64.2 + 17.2i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−10.78964914412335952065684975959, −10.05297577222771184568098782638, −8.817143389333080061452668991990, −8.325336974031626909192131502635, −6.88934902499798843098060106992, −5.63761306732714835126888793305, −4.48408917798117936062220316781, −3.57265013422337033734331350794, −1.97880252384278742705133107964, −0.22156420697015050367786911841, 2.37755498528542697382829607215, 4.02332793436250553759373833282, 4.79589425395880233235867020766, 6.21263016425121511723422407487, 7.17787195961130728154076810087, 7.88394916350271449385967790260, 8.740589497520768044086804495433, 10.37027341350672505907513614838, 10.62580296231448332951553099807, 11.91903963770754678782888565661

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