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Properties

Label 2-338-13.4-c3-0-33
Degree $2$
Conductor $338$
Sign $0.0841 + 0.996i$
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 + (0.622 − 1.07i)3-s + (1.99 + 3.46i)4-s − 20.5i·5-s + (2.15 − 1.24i)6-s + (18.4 − 10.6i)7-s + 7.99i·8-s + (12.7 + 22.0i)9-s + (20.5 − 35.5i)10-s + (−45.9 − 26.5i)11-s + 4.98·12-s + 42.5·14-s + (−22.1 − 12.7i)15-s + (−8 + 13.8i)16-s + (−34.6 − 59.9i)17-s + 50.8i·18-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.119 − 0.207i)3-s + (0.249 + 0.433i)4-s − 1.83i·5-s + (0.146 − 0.0847i)6-s + (0.993 − 0.573i)7-s + 0.353i·8-s + (0.471 + 0.816i)9-s + (0.648 − 1.12i)10-s + (−1.25 − 0.727i)11-s + 0.119·12-s + 0.811·14-s + (−0.380 − 0.219i)15-s + (−0.125 + 0.216i)16-s + (−0.493 − 0.855i)17-s + 0.666i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0841 + 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0841 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $0.0841 + 0.996i$
Analytic conductor: \(19.9426\)
Root analytic conductor: \(4.46571\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :3/2),\ 0.0841 + 0.996i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.679843078\)
\(L(\frac12)\) \(\approx\) \(2.679843078\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.622 + 1.07i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 20.5iT - 125T^{2} \)
7 \( 1 + (-18.4 + 10.6i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (45.9 + 26.5i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (34.6 + 59.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-39.8 + 23.0i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (43.6 - 75.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-81.0 + 140. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 28.3iT - 2.97e4T^{2} \)
37 \( 1 + (96.9 + 55.9i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-72.9 - 42.1i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (164. + 284. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 63.2iT - 1.03e5T^{2} \)
53 \( 1 - 721.T + 1.48e5T^{2} \)
59 \( 1 + (-709. + 409. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-198. - 344. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (67.4 + 38.9i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-624. + 360. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 57.1iT - 3.89e5T^{2} \)
79 \( 1 + 419.T + 4.93e5T^{2} \)
83 \( 1 - 917. iT - 5.71e5T^{2} \)
89 \( 1 + (328. + 189. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (300. - 173. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.15386940930845902546945051422, −9.958530472859566779553353352669, −8.612893949473297202781168138292, −8.036190373266643370288401685550, −7.31221906248212515468418099179, −5.44662039614328278535923451922, −5.03807172710048136000385529281, −4.13657438736487968517915652246, −2.18474614709589689505723538942, −0.75850515719202127173885847193, 1.99661022148926967086311631639, 2.93204603150945446695540407121, 4.09101468856274269720556848912, 5.33993645813843285430740969798, 6.49614208368045295120230608296, 7.30138394418491732743404718010, 8.445953302868345638747281993313, 10.03873149985349852398555174158, 10.40109520433923868808933617512, 11.30754623132320200797405910138

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