Properties

Label 2-338-13.10-c1-0-2
Degree $2$
Conductor $338$
Sign $-0.265 - 0.964i$
Analytic cond. $2.69894$
Root an. cond. $1.64284$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + 3i·5-s + (−0.866 − 0.499i)6-s + (2.59 + 1.5i)7-s + 0.999i·8-s + (1 − 1.73i)9-s + (−1.5 − 2.59i)10-s + 0.999·12-s − 3·14-s + (−2.59 + 1.5i)15-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)17-s + 2i·18-s + (−5.19 − 3i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s + 1.34i·5-s + (−0.353 − 0.204i)6-s + (0.981 + 0.566i)7-s + 0.353i·8-s + (0.333 − 0.577i)9-s + (−0.474 − 0.821i)10-s + 0.288·12-s − 0.801·14-s + (−0.670 + 0.387i)15-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s + 0.471i·18-s + (−1.19 − 0.688i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $-0.265 - 0.964i$
Analytic conductor: \(2.69894\)
Root analytic conductor: \(1.64284\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1/2),\ -0.265 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.712587 + 0.934852i\)
\(L(\frac12)\) \(\approx\) \(0.712587 + 0.934852i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 \)
good3 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3iT - 5T^{2} \)
7 \( 1 + (-2.59 - 1.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.19 + 3i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (2.59 - 1.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (5.19 + 3i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.3 + 6i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-12.9 - 7.5i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + (-5.19 + 3i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.3 + 6i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.31137170421061882767868157347, −10.88924392058307561950531551192, −9.927890620681850450133591627846, −8.999285369210166393998295798869, −8.162647544386054386160706964134, −7.02123053564600324647076705768, −6.29039010559846617350780282669, −4.89006542560326743576530482382, −3.46913893356173703166039287535, −2.07253797230842200141053444076, 1.07741076993337796819954356521, 2.20129135182188654155160173172, 4.26495084784979244511806812135, 5.03065078873181092952592066372, 6.75424439962145457817892771333, 7.88404178805534833917109313714, 8.361823819920976776212553359144, 9.210346826586658724974600675250, 10.44817688297070352352606071661, 11.13775196220787988774193373680

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