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Properties

Label 2-357-21.20-c1-0-24
Degree $2$
Conductor $357$
Sign $0.962 - 0.270i$
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  + 0.735i·2-s + (1.46 − 0.924i)3-s + 1.45·4-s − 2.26·5-s + (0.680 + 1.07i)6-s + (2.53 + 0.755i)7-s + 2.54i·8-s + (1.29 − 2.70i)9-s − 1.66i·10-s − 3.11i·11-s + (2.13 − 1.34i)12-s + 4.85i·13-s + (−0.555 + 1.86i)14-s + (−3.31 + 2.08i)15-s + 1.04·16-s + 17-s + ⋯
L(s)  = 1  + 0.520i·2-s + (0.845 − 0.533i)3-s + 0.729·4-s − 1.01·5-s + (0.277 + 0.439i)6-s + (0.958 + 0.285i)7-s + 0.899i·8-s + (0.430 − 0.902i)9-s − 0.525i·10-s − 0.938i·11-s + (0.616 − 0.389i)12-s + 1.34i·13-s + (−0.148 + 0.498i)14-s + (−0.854 + 0.539i)15-s + 0.261·16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $0.962 - 0.270i$
Analytic conductor: \(2.85065\)
Root analytic conductor: \(1.68838\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{357} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 357,\ (\ :1/2),\ 0.962 - 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92960 + 0.265591i\)
\(L(\frac12)\) \(\approx\) \(1.92960 + 0.265591i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.46 + 0.924i)T \)
7 \( 1 + (-2.53 - 0.755i)T \)
17 \( 1 - T \)
good2 \( 1 - 0.735iT - 2T^{2} \)
5 \( 1 + 2.26T + 5T^{2} \)
11 \( 1 + 3.11iT - 11T^{2} \)
13 \( 1 - 4.85iT - 13T^{2} \)
19 \( 1 + 4.59iT - 19T^{2} \)
23 \( 1 - 3.96iT - 23T^{2} \)
29 \( 1 + 1.95iT - 29T^{2} \)
31 \( 1 + 3.66iT - 31T^{2} \)
37 \( 1 + 9.03T + 37T^{2} \)
41 \( 1 - 8.70T + 41T^{2} \)
43 \( 1 + 8.87T + 43T^{2} \)
47 \( 1 + 5.25T + 47T^{2} \)
53 \( 1 - 11.2iT - 53T^{2} \)
59 \( 1 + 8.25T + 59T^{2} \)
61 \( 1 - 10.6iT - 61T^{2} \)
67 \( 1 + 1.26T + 67T^{2} \)
71 \( 1 + 13.1iT - 71T^{2} \)
73 \( 1 + 7.82iT - 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + 1.54T + 83T^{2} \)
89 \( 1 + 1.78T + 89T^{2} \)
97 \( 1 - 2.04iT - 97T^{2} \)
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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

−11.65800906331131238592386543710, −10.92898906520181461467464404741, −9.204556908654166001775178221165, −8.428406152657755170503034355604, −7.68481059733349003319671430053, −7.04340901782512691989764855950, −5.91522006075457948769581555958, −4.41395264632404242127801556154, −3.10051652344454562890326171197, −1.74441199169075200100869901210, 1.74854987021232838220162286798, 3.15451582245022004846079199406, 4.04084725896596433077547088677, 5.18284721439526991008159066323, 7.02768998105648588663491188812, 7.85800386684769178695525527458, 8.361837360509180943196967646404, 9.955808585193179218653978206864, 10.46043797775312221241826005216, 11.29162518594671287231685181759

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