Properties

Label 2-338-13.3-c3-0-13
Degree $2$
Conductor $338$
Sign $-0.0128 + 0.999i$
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 − 1.73i)2-s + (−2 − 3.46i)3-s + (−1.99 + 3.46i)4-s − 18·5-s + (−3.99 + 6.92i)6-s + (−10 + 17.3i)7-s + 7.99·8-s + (5.50 − 9.52i)9-s + (18 + 31.1i)10-s + (24 + 41.5i)11-s + 15.9·12-s + 40·14-s + (36 + 62.3i)15-s + (−8 − 13.8i)16-s + (−33 + 57.1i)17-s − 22·18-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.384 − 0.666i)3-s + (−0.249 + 0.433i)4-s − 1.60·5-s + (−0.272 + 0.471i)6-s + (−0.539 + 0.935i)7-s + 0.353·8-s + (0.203 − 0.352i)9-s + (0.569 + 0.985i)10-s + (0.657 + 1.13i)11-s + 0.384·12-s + 0.763·14-s + (0.619 + 1.07i)15-s + (−0.125 − 0.216i)16-s + (−0.470 + 0.815i)17-s − 0.288·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $-0.0128 + 0.999i$
Analytic conductor: \(19.9426\)
Root analytic conductor: \(4.46571\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :3/2),\ -0.0128 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5952136478\)
\(L(\frac12)\) \(\approx\) \(0.5952136478\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
13 \( 1 \)
good3 \( 1 + (2 + 3.46i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 18T + 125T^{2} \)
7 \( 1 + (10 - 17.3i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-24 - 41.5i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (33 - 57.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-8 + 13.8i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (84 + 145. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 20T + 2.97e4T^{2} \)
37 \( 1 + (127 + 219. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-195 - 337. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-62 + 107. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 468T + 1.03e5T^{2} \)
53 \( 1 - 558T + 1.48e5T^{2} \)
59 \( 1 + (-48 + 83.1i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-413 + 715. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-80 - 138. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-210 + 363. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 362T + 3.89e5T^{2} \)
79 \( 1 - 776T + 4.93e5T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + (813 + 1.40e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-647 + 1.12e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.21782759606127006318309868009, −9.973648641765421886781701531210, −8.976558111229072345918055801937, −8.102842478582627260086737971183, −7.10231972777536168992340381319, −6.29902244239417238411711653275, −4.50069135023256668013015384362, −3.62268077772743901137959722049, −2.07666952277646208466002238400, −0.43769635512455103228218936551, 0.69880204178286530275223249204, 3.58922201192660747473123659914, 4.17526339642412635099799215653, 5.39890127734916384538248571798, 6.77342476693903807043098489924, 7.52904831264959363218795114306, 8.382821401789467171793381100729, 9.491872697363616112831980572770, 10.44819065425930979156485156588, 11.27437239494547146492453416813

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