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Properties

Label 2-338-13.2-c2-0-22
Degree $2$
Conductor $338$
Sign $-0.955 + 0.295i$
Analytic cond. $9.20983$
Root an. cond. $3.03477$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 0.366i)2-s + (−1.52 − 2.63i)3-s + (1.73 + i)4-s + (4.79 − 4.79i)5-s + (1.11 + 4.15i)6-s + (−4.25 + 1.13i)7-s + (−1.99 − 2i)8-s + (−0.132 + 0.228i)9-s + (−8.29 + 4.79i)10-s + (3.71 − 13.8i)11-s − 6.08i·12-s + 6.22·14-s + (−19.9 − 5.33i)15-s + (1.99 + 3.46i)16-s + (20.9 + 12.1i)17-s + (0.264 − 0.264i)18-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.507 − 0.878i)3-s + (0.433 + 0.250i)4-s + (0.958 − 0.958i)5-s + (0.185 + 0.692i)6-s + (−0.607 + 0.162i)7-s + (−0.249 − 0.250i)8-s + (−0.0146 + 0.0254i)9-s + (−0.829 + 0.479i)10-s + (0.337 − 1.26i)11-s − 0.507i·12-s + 0.444·14-s + (−1.32 − 0.355i)15-s + (0.124 + 0.216i)16-s + (1.23 + 0.713i)17-s + (0.0146 − 0.0146i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $-0.955 + 0.295i$
Analytic conductor: \(9.20983\)
Root analytic conductor: \(3.03477\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1),\ -0.955 + 0.295i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.146314 - 0.966749i\)
\(L(\frac12)\) \(\approx\) \(0.146314 - 0.966749i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
13 \( 1 \)
good3 \( 1 + (1.52 + 2.63i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-4.79 + 4.79i)T - 25iT^{2} \)
7 \( 1 + (4.25 - 1.13i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (-3.71 + 13.8i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (-20.9 - 12.1i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (6.82 + 25.4i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (-5.44 + 3.14i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-11.1 - 19.2i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (8.59 - 8.59i)T - 961iT^{2} \)
37 \( 1 + (-1.64 + 6.13i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (70.4 + 18.8i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (26.9 + 15.5i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-7.65 - 7.65i)T + 2.20e3iT^{2} \)
53 \( 1 + 33.7T + 2.80e3T^{2} \)
59 \( 1 + (36.4 - 9.77i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-11.5 + 19.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-103. - 27.8i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (0.591 + 2.20i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (-38.1 - 38.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 19.1T + 6.24e3T^{2} \)
83 \( 1 + (-34.7 + 34.7i)T - 6.88e3iT^{2} \)
89 \( 1 + (0.930 - 3.47i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-6.51 - 24.3i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−10.91398805653238533863980004615, −9.839455466753752555436479163485, −9.024703035052849139291766785621, −8.310709196198614689184581233468, −6.90361624355589323642650365489, −6.17441992581326762949072513730, −5.29983383408071909629490286475, −3.30580146965920969624551059651, −1.62023698375056491919522570079, −0.61969679723281753601417921798, 1.91396438853635899204927284714, 3.43879177339645181365268348998, 4.96800167169986384396118289964, 6.05139967046622943152788105895, 6.86206731816248231746900948182, 7.889019611634294829185018109352, 9.564934170370923194852434311172, 9.957191560204935199937748335768, 10.30674714965415502925739730829, 11.43478599454712061608117243391

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