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Properties

Label 2-338-13.10-c3-0-4
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 + (−2.93 − 5.07i)3-s + (1.99 − 3.46i)4-s + 10.8i·5-s + (10.1 + 5.86i)6-s + (25.8 + 14.9i)7-s + 7.99i·8-s + (−3.70 + 6.41i)9-s + (−10.8 − 18.8i)10-s + (−42.4 + 24.5i)11-s − 23.4·12-s − 59.7·14-s + (55.1 − 31.8i)15-s + (−8 − 13.8i)16-s + (−3.03 + 5.26i)17-s − 14.8i·18-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.564 − 0.977i)3-s + (0.249 − 0.433i)4-s + 0.971i·5-s + (0.691 + 0.399i)6-s + (1.39 + 0.806i)7-s + 0.353i·8-s + (−0.137 + 0.237i)9-s + (−0.343 − 0.595i)10-s + (−1.16 + 0.672i)11-s − 0.564·12-s − 1.14·14-s + (0.950 − 0.548i)15-s + (−0.125 − 0.216i)16-s + (−0.0433 + 0.0750i)17-s − 0.193i·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

\(L(2)\) \(\approx\) \(0.6042972460\)
\(L(\frac12)\) \(\approx\) \(0.6042972460\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.93 + 5.07i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 10.8iT - 125T^{2} \)
7 \( 1 + (-25.8 - 14.9i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (42.4 - 24.5i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (3.03 - 5.26i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (5.07 + 2.93i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (2.79 + 4.84i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-5.30 - 9.19i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 316. iT - 2.97e4T^{2} \)
37 \( 1 + (329. - 190. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (232. - 134. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (115. - 199. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 524. iT - 1.03e5T^{2} \)
53 \( 1 + 274.T + 1.48e5T^{2} \)
59 \( 1 + (-123. - 71.4i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (281. - 487. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-448. + 258. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (125. + 72.2i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 201. iT - 3.89e5T^{2} \)
79 \( 1 - 26.4T + 4.93e5T^{2} \)
83 \( 1 - 1.14e3iT - 5.71e5T^{2} \)
89 \( 1 + (574. - 331. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (119. + 68.7i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.37690849673476749156409897922, −10.71223312194888490670458260373, −9.642771519617851776243009953700, −8.249900850935379421073010405436, −7.67489688508020098937801761552, −6.80798478120333860250745644543, −5.85912776532393590769076584373, −4.85933886771107384621629355604, −2.56869791771865697359717370465, −1.56438903509914293313333928009, 0.28402074319882111555355107042, 1.68513244868644053956339535474, 3.67154642900045509086949414952, 4.94161860972544974853277685720, 5.23961519272080087661580952661, 7.20440514524411438138712463491, 8.266993607448287654165908902305, 8.804599725498628315869561067805, 10.23017471616967562864269975196, 10.60294341017151400034548232749

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