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Properties

Label 2-338-13.3-c3-0-35
Degree $2$
Conductor $338$
Sign $-0.562 - 0.826i$
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 + (−1.67 − 2.90i)3-s + (−1.99 + 3.46i)4-s + 6.80·5-s + (−3.35 + 5.81i)6-s + (−7.76 + 13.4i)7-s + 7.99·8-s + (7.86 − 13.6i)9-s + (−6.80 − 11.7i)10-s + (−9.80 − 16.9i)11-s + 13.4·12-s + 31.0·14-s + (−11.4 − 19.7i)15-s + (−8 − 13.8i)16-s + (−62.9 + 109. i)17-s − 31.4·18-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.323 − 0.559i)3-s + (−0.249 + 0.433i)4-s + 0.608·5-s + (−0.228 + 0.395i)6-s + (−0.419 + 0.726i)7-s + 0.353·8-s + (0.291 − 0.504i)9-s + (−0.215 − 0.372i)10-s + (−0.268 − 0.465i)11-s + 0.323·12-s + 0.593·14-s + (−0.196 − 0.340i)15-s + (−0.125 − 0.216i)16-s + (−0.898 + 1.55i)17-s − 0.411·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $-0.562 - 0.826i$
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: \((2,\ 338,\ (\ :3/2),\ -0.562 - 0.826i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1101502278\)
\(L(\frac12)\) \(\approx\) \(0.1101502278\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.67 + 2.90i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 6.80T + 125T^{2} \)
7 \( 1 + (7.76 - 13.4i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (9.80 + 16.9i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (62.9 - 109. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-48.6 + 84.1i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (64.9 + 112. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (73.9 + 128. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 172.T + 2.97e4T^{2} \)
37 \( 1 + (-110. - 191. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-14.0 - 24.3i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (180. - 313. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 456.T + 1.03e5T^{2} \)
53 \( 1 + 643.T + 1.48e5T^{2} \)
59 \( 1 + (155. - 269. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (405. - 702. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (3.50 + 6.06i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-425. + 736. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 1.12e3T + 3.89e5T^{2} \)
79 \( 1 - 278.T + 4.93e5T^{2} \)
83 \( 1 + 417.T + 5.71e5T^{2} \)
89 \( 1 + (-162. - 280. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-41.0 + 71.0i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.55570420037882767777513225729, −9.542516349291478801454044487569, −8.874584864688726719661325050938, −7.75934521789105466092377294897, −6.41727060226573768031153113172, −5.85988459107133102787513502323, −4.21885308652593982300053284370, −2.75819790416624013943584401189, −1.61285183821193435692557647651, −0.04414515973598885445615711948, 1.87286468899955480061164888706, 3.80626594586676899720621004984, 5.00987581563364782378873945088, 5.78408717750162106831263671940, 7.13292105080760607778089471467, 7.65343855409383640895052670045, 9.260422008775249771007609980278, 9.763633652341901251984564124986, 10.52516872485987633301080756559, 11.42055664625180150492615943610

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