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Properties

Label 2-338-13.4-c3-0-19
Degree $2$
Conductor $338$
Sign $0.327 - 0.944i$
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 + (−1.67 + 2.90i)3-s + (1.99 + 3.46i)4-s − 6.80i·5-s + (−5.81 + 3.35i)6-s + (13.4 − 7.76i)7-s + 7.99i·8-s + (7.86 + 13.6i)9-s + (6.80 − 11.7i)10-s + (−16.9 − 9.80i)11-s − 13.4·12-s + 31.0·14-s + (19.7 + 11.4i)15-s + (−8 + 13.8i)16-s + (62.9 + 109. i)17-s + 31.4i·18-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.323 + 0.559i)3-s + (0.249 + 0.433i)4-s − 0.608i·5-s + (−0.395 + 0.228i)6-s + (0.726 − 0.419i)7-s + 0.353i·8-s + (0.291 + 0.504i)9-s + (0.215 − 0.372i)10-s + (−0.465 − 0.268i)11-s − 0.323·12-s + 0.593·14-s + (0.340 + 0.196i)15-s + (−0.125 + 0.216i)16-s + (0.898 + 1.55i)17-s + 0.411i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $0.327 - 0.944i$
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: \((2,\ 338,\ (\ :3/2),\ 0.327 - 0.944i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.681874100\)
\(L(\frac12)\) \(\approx\) \(2.681874100\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.67 - 2.90i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 6.80iT - 125T^{2} \)
7 \( 1 + (-13.4 + 7.76i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (16.9 + 9.80i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-62.9 - 109. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-84.1 + 48.6i)T + (3.42e3 - 5.94e3i)T^{2} \)
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 \( 1 - 172. iT - 2.97e4T^{2} \)
37 \( 1 + (-191. - 110. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (24.3 + 14.0i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-180. - 313. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 456. iT - 1.03e5T^{2} \)
53 \( 1 + 643.T + 1.48e5T^{2} \)
59 \( 1 + (-269. + 155. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (405. + 702. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-6.06 - 3.50i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-736. + 425. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 1.12e3iT - 3.89e5T^{2} \)
79 \( 1 - 278.T + 4.93e5T^{2} \)
83 \( 1 - 417. iT - 5.71e5T^{2} \)
89 \( 1 + (-280. - 162. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-71.0 + 41.0i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.01566746393021574944688777372, −10.76969387573599107564824999409, −9.461127340459413334730313173173, −8.225064051695225223232055007670, −7.60768604941582533189743618566, −6.21020874832795463732895511413, −4.98859613041482958180860376489, −4.68492105100075040462294568303, −3.25629251016813230089408409566, −1.35229409589158269901490038006, 0.962249725818217175479211349815, 2.38676607133104433079549870171, 3.59428450099678319233873907574, 5.10345411777553571916590782244, 5.83059124333673250204245430200, 7.15720978689615040636471966046, 7.65737349148508557176558189504, 9.340543176082677203312956612608, 10.09655106469691579910482462853, 11.49836030831675194919277766192

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