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Properties

Label 2-351-13.10-c1-0-17
Degree $2$
Conductor $351$
Sign $-0.252 + 0.967i$
Analytic cond. $2.80274$
Root an. cond. $1.67414$
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  + (1.5 − 0.866i)2-s + (0.5 − 0.866i)4-s − 3.46i·5-s + (−3 − 1.73i)7-s + 1.73i·8-s + (−2.99 − 5.19i)10-s + (3 − 1.73i)11-s + (−1 − 3.46i)13-s − 6·14-s + (2.49 + 4.33i)16-s + (4.5 + 2.59i)19-s + (−3.00 − 1.73i)20-s + (3 − 5.19i)22-s + (3 + 5.19i)23-s − 6.99·25-s + (−4.5 − 4.33i)26-s + ⋯
L(s)  = 1  + (1.06 − 0.612i)2-s + (0.250 − 0.433i)4-s − 1.54i·5-s + (−1.13 − 0.654i)7-s + 0.612i·8-s + (−0.948 − 1.64i)10-s + (0.904 − 0.522i)11-s + (−0.277 − 0.960i)13-s − 1.60·14-s + (0.624 + 1.08i)16-s + (1.03 + 0.596i)19-s + (−0.670 − 0.387i)20-s + (0.639 − 1.10i)22-s + (0.625 + 1.08i)23-s − 1.39·25-s + (−0.882 − 0.849i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $-0.252 + 0.967i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ -0.252 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23548 - 1.59946i\)
\(L(\frac12)\) \(\approx\) \(1.23548 - 1.59946i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (1 + 3.46i)T \)
good2 \( 1 + (-1.5 + 0.866i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 3.46iT - 5T^{2} \)
7 \( 1 + (3 + 1.73i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.5 - 2.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.19iT - 31T^{2} \)
37 \( 1 + (7.5 - 4.33i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6 + 3.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.5 + 6.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-6 - 3.46i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.5 - 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.5 - 0.866i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6 - 3.46i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 8.66iT - 73T^{2} \)
79 \( 1 - T + 79T^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 + (-9 + 5.19i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.5 + 0.866i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.64720704550957830591445260080, −10.40822225172739405858477461580, −9.402995957132381990193998829045, −8.584084570320579804891165261820, −7.39140053156027753139760172890, −5.86429829896778347299214252018, −5.13939630126606208148460198184, −3.95501615780070521541373020434, −3.21248480641982987197975637336, −1.12645353138182631956768735643, 2.68680651504493127098857211728, 3.61013153133796548668442463208, 4.84197668234992044224781161738, 6.35379851727281110462379709992, 6.53537046751859566160233476578, 7.35978622976922417125049713784, 9.255830403584461810449949987784, 9.777027577927745090925776642514, 10.95097245270519936623559866792, 12.02613386269436706790968756282

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