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Properties

Label 2-351-117.22-c1-0-9
Degree $2$
Conductor $351$
Sign $0.989 - 0.145i$
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  + 2.65·2-s + 5.05·4-s + (−0.324 + 0.561i)5-s + (−0.773 + 1.34i)7-s + 8.11·8-s + (−0.861 + 1.49i)10-s − 5.07·11-s + (−0.445 − 3.57i)13-s + (−2.05 + 3.56i)14-s + 11.4·16-s + (0.103 + 0.179i)17-s + (−1.79 − 3.10i)19-s + (−1.63 + 2.84i)20-s − 13.4·22-s + (−1.60 − 2.77i)23-s + ⋯
L(s)  = 1  + 1.87·2-s + 2.52·4-s + (−0.145 + 0.251i)5-s + (−0.292 + 0.506i)7-s + 2.86·8-s + (−0.272 + 0.471i)10-s − 1.52·11-s + (−0.123 − 0.992i)13-s + (−0.549 + 0.951i)14-s + 2.86·16-s + (0.0251 + 0.0436i)17-s + (−0.411 − 0.713i)19-s + (−0.366 + 0.635i)20-s − 2.87·22-s + (−0.333 − 0.578i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $0.989 - 0.145i$
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: \((2,\ 351,\ (\ :1/2),\ 0.989 - 0.145i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.50294 + 0.256947i\)
\(L(\frac12)\) \(\approx\) \(3.50294 + 0.256947i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.445 + 3.57i)T \)
good2 \( 1 - 2.65T + 2T^{2} \)
5 \( 1 + (0.324 - 0.561i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.773 - 1.34i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 5.07T + 11T^{2} \)
17 \( 1 + (-0.103 - 0.179i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.79 + 3.10i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.60 + 2.77i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.83T + 29T^{2} \)
31 \( 1 + (-1.58 + 2.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.71 - 8.16i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.30 + 7.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.99 - 5.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.42 - 2.47i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.48T + 53T^{2} \)
59 \( 1 - 2.98T + 59T^{2} \)
61 \( 1 + (4.02 - 6.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.47 + 4.28i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.787 - 1.36i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.03T + 73T^{2} \)
79 \( 1 + (-3.23 - 5.60i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.24 + 2.15i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.76 + 3.05i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.69 + 8.12i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.85384947892262497514965889416, −10.78438117434610649349425137791, −10.23647997490535502436809948165, −8.383391057522815434850340678022, −7.35251525349053067027229969586, −6.34958473722685175886149289437, −5.40120242277655537479055908766, −4.67326347644742149860713989399, −3.16314402864454995262157977352, −2.53985507527979143280938304461, 2.19130491895488574946516365393, 3.44998432716397151898763526622, 4.49209285506940854681326062961, 5.30095316649328098183706259481, 6.41621024023740924268449715183, 7.25761672131073497197376424247, 8.336029713329511338001316307075, 10.13648963908755417045099064009, 10.74826974092493917171175766545, 11.90768042218623563061874560671

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