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Properties

Label 2-357-21.20-c1-0-18
Degree $2$
Conductor $357$
Sign $0.932 - 0.361i$
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  + 0.889i·2-s + (0.351 − 1.69i)3-s + 1.20·4-s + 1.21·5-s + (1.50 + 0.312i)6-s + (1.43 + 2.22i)7-s + 2.85i·8-s + (−2.75 − 1.19i)9-s + 1.08i·10-s + 5.85i·11-s + (0.424 − 2.05i)12-s − 5.22i·13-s + (−1.97 + 1.27i)14-s + (0.426 − 2.06i)15-s − 0.117·16-s + 17-s + ⋯
L(s)  = 1  + 0.628i·2-s + (0.202 − 0.979i)3-s + 0.604·4-s + 0.543·5-s + (0.615 + 0.127i)6-s + (0.543 + 0.839i)7-s + 1.00i·8-s + (−0.917 − 0.397i)9-s + 0.341i·10-s + 1.76i·11-s + (0.122 − 0.592i)12-s − 1.44i·13-s + (−0.527 + 0.341i)14-s + (0.110 − 0.532i)15-s − 0.0294·16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 - 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $0.932 - 0.361i$
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: \((2,\ 357,\ (\ :1/2),\ 0.932 - 0.361i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81851 + 0.340151i\)
\(L(\frac12)\) \(\approx\) \(1.81851 + 0.340151i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.351 + 1.69i)T \)
7 \( 1 + (-1.43 - 2.22i)T \)
17 \( 1 - T \)
good2 \( 1 - 0.889iT - 2T^{2} \)
5 \( 1 - 1.21T + 5T^{2} \)
11 \( 1 - 5.85iT - 11T^{2} \)
13 \( 1 + 5.22iT - 13T^{2} \)
19 \( 1 + 0.437iT - 19T^{2} \)
23 \( 1 + 7.36iT - 23T^{2} \)
29 \( 1 + 3.77iT - 29T^{2} \)
31 \( 1 + 1.75iT - 31T^{2} \)
37 \( 1 - 5.74T + 37T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 - 2.95T + 43T^{2} \)
47 \( 1 + 4.88T + 47T^{2} \)
53 \( 1 - 0.530iT - 53T^{2} \)
59 \( 1 - 0.268T + 59T^{2} \)
61 \( 1 + 7.68iT - 61T^{2} \)
67 \( 1 + 8.07T + 67T^{2} \)
71 \( 1 + 4.66iT - 71T^{2} \)
73 \( 1 - 6.01iT - 73T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 - 10.0T + 83T^{2} \)
89 \( 1 + 9.27T + 89T^{2} \)
97 \( 1 - 11.7iT - 97T^{2} \)
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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

−11.81417904240923804281867187887, −10.61875912725019068950564799445, −9.555513109671742443735937541661, −8.227384858583128412830029179890, −7.79265023433024120731353204400, −6.72460993780559949323154360296, −5.91273039969818768694313718367, −5.00554798765920718012372060662, −2.65842882262567949445701896186, −1.90430397492364927671975600999, 1.61316102996726409399010165007, 3.19822132852735495995397829667, 4.00859513218894836975051826761, 5.43103113815979202898257091489, 6.47687900255299750868826878646, 7.75589607989149419973724512947, 8.906878861305230408235356620319, 9.781746912379781519153941415729, 10.57921602837862832401844047032, 11.36730384612176801753830726449

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