Properties

Label 2-338-13.10-c3-0-20
Degree $2$
Conductor $338$
Sign $0.711 + 0.702i$
Analytic cond. $19.9426$
Root an. cond. $4.46571$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.73 + i)2-s + (1.5 + 2.59i)3-s + (1.99 − 3.46i)4-s + 2i·5-s + (−5.19 − 3i)6-s + (−4.33 − 2.5i)7-s + 7.99i·8-s + (9 − 15.5i)9-s + (−2 − 3.46i)10-s + (−11.2 + 6.5i)11-s + 12·12-s + 10·14-s + (−5.19 + 3i)15-s + (−8 − 13.8i)16-s + (13.5 − 23.3i)17-s + 36i·18-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s + 0.178i·5-s + (−0.353 − 0.204i)6-s + (−0.233 − 0.134i)7-s + 0.353i·8-s + (0.333 − 0.577i)9-s + (−0.0632 − 0.109i)10-s + (−0.308 + 0.178i)11-s + 0.288·12-s + 0.190·14-s + (−0.0894 + 0.0516i)15-s + (−0.125 − 0.216i)16-s + (0.192 − 0.333i)17-s + 0.471i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(19.9426\)
Root analytic conductor: \(4.46571\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :3/2),\ 0.711 + 0.702i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.109858650\)
\(L(\frac12)\) \(\approx\) \(1.109858650\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.73 - i)T \)
13 \( 1 \)
good3 \( 1 + (-1.5 - 2.59i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 2iT - 125T^{2} \)
7 \( 1 + (4.33 + 2.5i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (11.2 - 6.5i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-13.5 + 23.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (64.9 + 37.5i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (93.5 + 161. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-6.5 - 11.2i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 104iT - 2.97e4T^{2} \)
37 \( 1 + (366. - 211.5i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-168. + 97.5i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-99.5 + 172. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 388iT - 1.03e5T^{2} \)
53 \( 1 - 618T + 1.48e5T^{2} \)
59 \( 1 + (-425. - 245.5i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (87.5 - 151. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-707. + 408.5i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (68.4 + 39.5i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 230iT - 3.89e5T^{2} \)
79 \( 1 - 764T + 4.93e5T^{2} \)
83 \( 1 + 732iT - 5.71e5T^{2} \)
89 \( 1 + (-901. + 520.5i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-84.0 - 48.5i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−10.46660749712668467598503916624, −10.20118173486377738199417029584, −9.039575370052370749364448702081, −8.413632882672585343612439975875, −7.09680936609931681800669605365, −6.43689690021670849044067944277, −5.01331986607467724357860453184, −3.81694874899143257333815126966, −2.39365162547575504904252701088, −0.49111635197951546144852164503, 1.33224820165570503825058129414, 2.47637496989135140138993113969, 3.85140955688531503597388391680, 5.35689959665070933064717524175, 6.65240965997560669884921211926, 7.68393281538988205454811988069, 8.348107937741202867812097461602, 9.343465137297569504155261479140, 10.34138209806316799696322537584, 11.04043811219822478130281322799

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