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Properties

Label 2-338-13.3-c1-0-6
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 + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + 3·5-s + (−0.499 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (1 − 1.73i)9-s + (−1.5 − 2.59i)10-s + (3 + 5.19i)11-s + 0.999·12-s + 0.999·14-s + (−1.5 − 2.59i)15-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)17-s − 2·18-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + 1.34·5-s + (−0.204 + 0.353i)6-s + (−0.188 + 0.327i)7-s + 0.353·8-s + (0.333 − 0.577i)9-s + (−0.474 − 0.821i)10-s + (0.904 + 1.56i)11-s + 0.288·12-s + 0.267·14-s + (−0.387 − 0.670i)15-s + (−0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s − 0.471·18-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

\(L(1)\) \(\approx\) \(1.12269 - 0.629133i\)
\(L(\frac12)\) \(\approx\) \(1.12269 - 0.629133i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3T + 5T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7 - 12.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.5 - 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5 - 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.60967619699366416382551611675, −10.19873472295498509785689001808, −9.572584113146321138249905525998, −9.101405691148172230036869748750, −7.42364860981592227254693409780, −6.63631819452094475795831718184, −5.59186003148210604003732412198, −4.20393011018552674614268859520, −2.46865455591262811545435441123, −1.39614105227617374038853612553, 1.51527637428858548459108106953, 3.55887349022475377528249598166, 5.06130526409527960770800136433, 5.92134663573667486756704022253, 6.62572093069655918811370948820, 8.026027357253440522006233339658, 9.001602777616016471459907645130, 9.864935025021312356477139897005, 10.49186441281234235838314060196, 11.35543051478638778090258748762

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