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Properties

Label 2-338-13.3-c3-0-33
Degree $2$
Conductor $338$
Sign $-0.872 + 0.488i$
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 + 1.73i)2-s + (−2.93 − 5.07i)3-s + (−1.99 + 3.46i)4-s + 10.8·5-s + (5.86 − 10.1i)6-s + (−14.9 + 25.8i)7-s − 7.99·8-s + (−3.70 + 6.41i)9-s + (10.8 + 18.8i)10-s + (−24.5 − 42.4i)11-s + 23.4·12-s − 59.7·14-s + (−31.8 − 55.1i)15-s + (−8 − 13.8i)16-s + (3.03 − 5.26i)17-s − 14.8·18-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.564 − 0.977i)3-s + (−0.249 + 0.433i)4-s + 0.971·5-s + (0.399 − 0.691i)6-s + (−0.806 + 1.39i)7-s − 0.353·8-s + (−0.137 + 0.237i)9-s + (0.343 + 0.595i)10-s + (−0.672 − 1.16i)11-s + 0.564·12-s − 1.14·14-s + (−0.548 − 0.950i)15-s + (−0.125 − 0.216i)16-s + (0.0433 − 0.0750i)17-s − 0.193·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $-0.872 + 0.488i$
Analytic conductor: \(19.9426\)
Root analytic conductor: \(4.46571\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :3/2),\ -0.872 + 0.488i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2380969588\)
\(L(\frac12)\) \(\approx\) \(0.2380969588\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
13 \( 1 \)
good3 \( 1 + (2.93 + 5.07i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 10.8T + 125T^{2} \)
7 \( 1 + (14.9 - 25.8i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (24.5 + 42.4i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-3.03 + 5.26i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (2.93 - 5.07i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-2.79 - 4.84i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-5.30 - 9.19i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 316.T + 2.97e4T^{2} \)
37 \( 1 + (190. + 329. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-134. - 232. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-115. + 199. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 524.T + 1.03e5T^{2} \)
53 \( 1 + 274.T + 1.48e5T^{2} \)
59 \( 1 + (71.4 - 123. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (281. - 487. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (258. + 448. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (72.2 - 125. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 201.T + 3.89e5T^{2} \)
79 \( 1 - 26.4T + 4.93e5T^{2} \)
83 \( 1 - 1.14e3T + 5.71e5T^{2} \)
89 \( 1 + (331. + 574. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (68.7 - 119. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.91220472585469329144429544251, −9.519380819766701767328654019093, −8.843923490631660202295890088219, −7.64152286519531170867674569401, −6.50873849590743717405456883487, −5.81633801972284012353740012102, −5.45237054727967764258885953150, −3.27713157306936143394590979460, −2.01663497500217691048363983357, −0.07434961302924994174188643875, 1.81684475025820271169101308142, 3.43622758298676762927042929691, 4.48842527074818064723364852967, 5.28511266468266724697675899643, 6.44629408110649901874504304600, 7.54866616895277878277750652037, 9.427355842063341059790103251957, 9.919033881766998275637767289475, 10.45950858151188458253466587560, 11.13918090758007332818980474374

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