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Properties

Label 2-338-13.3-c1-0-5
Degree $2$
Conductor $338$
Sign $-0.522 - 0.852i$
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  + (0.5 + 0.866i)2-s + (1.5 + 2.59i)3-s + (−0.499 + 0.866i)4-s + 5-s + (−1.5 + 2.59i)6-s + (0.5 − 0.866i)7-s − 0.999·8-s + (−3 + 5.19i)9-s + (0.5 + 0.866i)10-s + (−1 − 1.73i)11-s − 3·12-s + 0.999·14-s + (1.5 + 2.59i)15-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)17-s − 6·18-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.866 + 1.49i)3-s + (−0.249 + 0.433i)4-s + 0.447·5-s + (−0.612 + 1.06i)6-s + (0.188 − 0.327i)7-s − 0.353·8-s + (−1 + 1.73i)9-s + (0.158 + 0.273i)10-s + (−0.301 − 0.522i)11-s − 0.866·12-s + 0.267·14-s + (0.387 + 0.670i)15-s + (−0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s − 1.41·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $-0.522 - 0.852i$
Analytic conductor: \(2.69894\)
Root analytic conductor: \(1.64284\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1/2),\ -0.522 - 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02078 + 1.82160i\)
\(L(\frac12)\) \(\approx\) \(1.02078 + 1.82160i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 \)
good3 \( 1 + (-1.5 - 2.59i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - T + 5T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.5 + 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 13T + 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.5 - 4.33i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7 + 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.64355721635335982740632901120, −10.77899740582499712534371250157, −9.704886083943442512656515031659, −9.209112803371729887947257938045, −8.192547814175295956492418782356, −7.23163919789772636519910985104, −5.62261035511708516711623105950, −4.89321223784464746283057464781, −3.76962507835140844885983781499, −2.78493287904463981493756497039, 1.50951539290733269938389551045, 2.38435336386728289082888716944, 3.63021027940924732879525151077, 5.40768158342631322031597599700, 6.38893399675427216344801348518, 7.53667461946666212315676486900, 8.319660007028431413684904528859, 9.337977912137359312454990147230, 10.28165707151217771111895206168, 11.59894487277404434885648120520

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