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Properties

Label 2-357-21.17-c1-0-27
Degree $2$
Conductor $357$
Sign $0.744 + 0.667i$
Analytic cond. $2.85065$
Root an. cond. $1.68838$
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 + (1.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (−1.5 + 2.59i)6-s + (2 − 1.73i)7-s − 1.73i·8-s + (1.5 − 2.59i)9-s + (−4.5 − 2.59i)11-s − 1.73i·12-s − 1.73i·13-s + (−1.50 + 4.33i)14-s + (2.49 + 4.33i)16-s + (0.5 − 0.866i)17-s + 5.19i·18-s + (−3 + 1.73i)19-s + ⋯
L(s)  = 1  + (−1.06 + 0.612i)2-s + (0.866 − 0.499i)3-s + (0.250 − 0.433i)4-s + (−0.612 + 1.06i)6-s + (0.755 − 0.654i)7-s − 0.612i·8-s + (0.5 − 0.866i)9-s + (−1.35 − 0.783i)11-s − 0.500i·12-s − 0.480i·13-s + (−0.400 + 1.15i)14-s + (0.624 + 1.08i)16-s + (0.121 − 0.210i)17-s + 1.22i·18-s + (−0.688 + 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $0.744 + 0.667i$
Analytic conductor: \(2.85065\)
Root analytic conductor: \(1.68838\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{357} (290, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 357,\ (\ :1/2),\ 0.744 + 0.667i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.904287 - 0.345714i\)
\(L(\frac12)\) \(\approx\) \(0.904287 - 0.345714i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 + (-2 + 1.73i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (1.5 - 0.866i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.5 + 2.59i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
19 \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 + 1.73i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 - 0.866i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.66iT - 71T^{2} \)
73 \( 1 + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.3iT - 97T^{2} \)
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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

−10.84069656271614907514045920784, −10.38550206941248187390460059652, −9.078052811650550077803277961258, −8.432101753239456614753089194595, −7.70386726548667629382258311519, −7.16329675363382277913775174685, −5.81643438844293537432422462577, −4.17951204814936996391668341798, −2.74636933825050812033665550066, −0.891209991267375494676414109742, 1.91153765527089265877366285493, 2.71877816493185534500829320180, 4.53316238507463371098917588586, 5.43107558942279330249189030798, 7.46250735230206576596703436842, 8.081162343787057476491695969995, 9.032345307381398114006130457934, 9.520578435676002546047974729243, 10.65283677522512738564764626395, 11.02212904422087574109059446505

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