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Properties

Label 2-357-21.20-c1-0-20
Degree $2$
Conductor $357$
Sign $0.785 + 0.618i$
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  + 1.84i·2-s + (−1.59 + 0.682i)3-s − 1.42·4-s − 2.40·5-s + (−1.26 − 2.94i)6-s + (−1.26 − 2.32i)7-s + 1.07i·8-s + (2.06 − 2.17i)9-s − 4.43i·10-s − 3.68i·11-s + (2.26 − 0.969i)12-s + 2.24i·13-s + (4.29 − 2.34i)14-s + (3.82 − 1.63i)15-s − 4.82·16-s + 17-s + ⋯
L(s)  = 1  + 1.30i·2-s + (−0.919 + 0.394i)3-s − 0.710·4-s − 1.07·5-s + (−0.515 − 1.20i)6-s + (−0.478 − 0.878i)7-s + 0.379i·8-s + (0.689 − 0.724i)9-s − 1.40i·10-s − 1.10i·11-s + (0.652 − 0.279i)12-s + 0.622i·13-s + (1.14 − 0.625i)14-s + (0.986 − 0.423i)15-s − 1.20·16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 + 0.618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.785 + 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $0.785 + 0.618i$
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: \((2,\ 357,\ (\ :1/2),\ 0.785 + 0.618i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.286062 - 0.0991116i\)
\(L(\frac12)\) \(\approx\) \(0.286062 - 0.0991116i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.59 - 0.682i)T \)
7 \( 1 + (1.26 + 2.32i)T \)
17 \( 1 - T \)
good2 \( 1 - 1.84iT - 2T^{2} \)
5 \( 1 + 2.40T + 5T^{2} \)
11 \( 1 + 3.68iT - 11T^{2} \)
13 \( 1 - 2.24iT - 13T^{2} \)
19 \( 1 + 4.04iT - 19T^{2} \)
23 \( 1 + 0.471iT - 23T^{2} \)
29 \( 1 + 1.76iT - 29T^{2} \)
31 \( 1 + 5.78iT - 31T^{2} \)
37 \( 1 + 11.0T + 37T^{2} \)
41 \( 1 + 7.07T + 41T^{2} \)
43 \( 1 - 3.80T + 43T^{2} \)
47 \( 1 - 9.80T + 47T^{2} \)
53 \( 1 + 13.4iT - 53T^{2} \)
59 \( 1 + 8.69T + 59T^{2} \)
61 \( 1 + 4.34iT - 61T^{2} \)
67 \( 1 + 12.7T + 67T^{2} \)
71 \( 1 - 4.79iT - 71T^{2} \)
73 \( 1 - 4.36iT - 73T^{2} \)
79 \( 1 - 13.7T + 79T^{2} \)
83 \( 1 + 7.25T + 83T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 - 0.875iT - 97T^{2} \)
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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

−11.32752256589622939139541184896, −10.62820523449257487156361322462, −9.330548547925647568772563619770, −8.286746855103560262021794267207, −7.26315956264416894158321081580, −6.67279802055152919531618336679, −5.68759196152239413776341079545, −4.55762312146611598592572737811, −3.61375800715521234446459826262, −0.23199994682243059839310006498, 1.66981423291780553826170297527, 3.14502768528119614697195196180, 4.32976118418626443041870373750, 5.54111391638344406746154233181, 6.83748472167763283223077836342, 7.71154189526804294648376798791, 9.040065537096812799031516788735, 10.22586107883040811649906864169, 10.68329122583483213196131750900, 11.88128386958201722115244763104

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