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Properties

Label 2-338-13.4-c3-0-32
Degree $2$
Conductor $338$
Sign $-0.978 - 0.207i$
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 + (−0.202 + 0.351i)3-s + (1.99 + 3.46i)4-s − 6.36i·5-s + (0.702 − 0.405i)6-s + (−2.20 + 1.27i)7-s − 7.99i·8-s + (13.4 + 23.2i)9-s + (−6.36 + 11.0i)10-s + (−22.6 − 13.0i)11-s − 1.62·12-s + 5.10·14-s + (2.23 + 1.29i)15-s + (−8 + 13.8i)16-s + (−46.8 − 81.1i)17-s − 53.6i·18-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.0390 + 0.0676i)3-s + (0.249 + 0.433i)4-s − 0.569i·5-s + (0.0478 − 0.0276i)6-s + (−0.119 + 0.0688i)7-s − 0.353i·8-s + (0.496 + 0.860i)9-s + (−0.201 + 0.348i)10-s + (−0.620 − 0.357i)11-s − 0.0390·12-s + 0.0973·14-s + (0.0384 + 0.0222i)15-s + (−0.125 + 0.216i)16-s + (−0.668 − 1.15i)17-s − 0.702i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.207i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $-0.978 - 0.207i$
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: \((2,\ 338,\ (\ :3/2),\ -0.978 - 0.207i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1719298671\)
\(L(\frac12)\) \(\approx\) \(0.1719298671\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.202 - 0.351i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 6.36iT - 125T^{2} \)
7 \( 1 + (2.20 - 1.27i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (22.6 + 13.0i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (46.8 + 81.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-32.2 + 18.6i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (52.4 - 90.8i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (124. - 216. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 278. iT - 2.97e4T^{2} \)
37 \( 1 + (-9.43 - 5.45i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (321. + 185. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (206. + 358. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 238. iT - 1.03e5T^{2} \)
53 \( 1 + 424.T + 1.48e5T^{2} \)
59 \( 1 + (670. - 387. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-61.7 - 106. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (763. + 440. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (102. - 59.3i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 209. iT - 3.89e5T^{2} \)
79 \( 1 + 532.T + 4.93e5T^{2} \)
83 \( 1 - 376. iT - 5.71e5T^{2} \)
89 \( 1 + (36.9 + 21.3i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (553. - 319. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−10.65806177800329369280697919977, −9.620091094079445647982478740671, −8.906077624853091578543913948776, −7.83457770431815100711643462173, −7.09479644484032353683100621637, −5.52276190185409159991818365410, −4.57422993562439570415399552299, −3.03000248180092872780894719604, −1.68115457801384361916567776499, −0.07303609752524847649026298262, 1.69789291561701999109720965013, 3.25961829442592949411396701286, 4.67998326115664461177905174740, 6.19667191462973407556084609596, 6.76128620214845374942417424214, 7.82793336704261732622310828214, 8.750270784937242225844225049007, 9.899394494387568659052652750907, 10.40168979664307558985193189943, 11.43429924126071776454059155976

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