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Properties

Label 2-351-117.22-c1-0-10
Degree $2$
Conductor $351$
Sign $-0.265 + 0.964i$
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  + 0.163·2-s − 1.97·4-s + (1.55 − 2.69i)5-s + (0.0682 − 0.118i)7-s − 0.649·8-s + (0.254 − 0.440i)10-s − 4.17·11-s + (0.300 − 3.59i)13-s + (0.0111 − 0.0193i)14-s + 3.84·16-s + (−2.67 − 4.64i)17-s + (−0.154 − 0.268i)19-s + (−3.07 + 5.32i)20-s − 0.682·22-s + (−0.961 − 1.66i)23-s + ⋯
L(s)  = 1  + 0.115·2-s − 0.986·4-s + (0.696 − 1.20i)5-s + (0.0257 − 0.0446i)7-s − 0.229·8-s + (0.0804 − 0.139i)10-s − 1.25·11-s + (0.0834 − 0.996i)13-s + (0.00297 − 0.00515i)14-s + 0.960·16-s + (−0.649 − 1.12i)17-s + (−0.0355 − 0.0614i)19-s + (−0.687 + 1.19i)20-s − 0.145·22-s + (−0.200 − 0.347i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $-0.265 + 0.964i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (334, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ -0.265 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.598996 - 0.786402i\)
\(L(\frac12)\) \(\approx\) \(0.598996 - 0.786402i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.300 + 3.59i)T \)
good2 \( 1 - 0.163T + 2T^{2} \)
5 \( 1 + (-1.55 + 2.69i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.0682 + 0.118i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 4.17T + 11T^{2} \)
17 \( 1 + (2.67 + 4.64i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.154 + 0.268i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.961 + 1.66i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.75T + 29T^{2} \)
31 \( 1 + (-2.28 + 3.95i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.48 + 7.77i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.56 - 6.18i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.32 - 9.21i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.663 - 1.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.10T + 53T^{2} \)
59 \( 1 - 4.80T + 59T^{2} \)
61 \( 1 + (-3.61 + 6.26i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.53 - 11.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.24 + 3.89i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.18T + 73T^{2} \)
79 \( 1 + (1.41 + 2.45i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.19 - 8.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.17 - 10.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.05 + 13.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.17216452186386765810749502463, −9.994004812142229939189994161450, −9.426791115023710894971807582454, −8.455476834159223063919637638867, −7.76732007891826412251966276523, −5.97529428812284136795518422423, −5.13175336185691459143211476376, −4.49463757356381672714832594606, −2.72469412368075090432656128789, −0.67378919830943866153223492279, 2.21263381492637791696000968303, 3.55497677477971194870404606935, 4.82949353898067388635340241443, 5.94057010180059787784864071788, 6.84418120932195813623386874433, 8.108544403869350802043909872333, 8.999256303850497300796554733956, 10.18305769494128225682516356901, 10.47202915318100770289104899617, 11.71038804273501698432465412425

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