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Properties

Label 2-354-3.2-c2-0-36
Degree $2$
Conductor $354$
Sign $-0.830 + 0.556i$
Analytic cond. $9.64580$
Root an. cond. $3.10576$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s + (2.49 − 1.67i)3-s − 2.00·4-s − 7.29i·5-s + (−2.36 − 3.52i)6-s + 6.28·7-s + 2.82i·8-s + (3.41 − 8.32i)9-s − 10.3·10-s − 1.79i·11-s + (−4.98 + 3.34i)12-s + 5.41·13-s − 8.88i·14-s + (−12.1 − 18.1i)15-s + 4.00·16-s + 3.95i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.830 − 0.556i)3-s − 0.500·4-s − 1.45i·5-s + (−0.393 − 0.587i)6-s + 0.897·7-s + 0.353i·8-s + (0.379 − 0.925i)9-s − 1.03·10-s − 0.163i·11-s + (−0.415 + 0.278i)12-s + 0.416·13-s − 0.634i·14-s + (−0.812 − 1.21i)15-s + 0.250·16-s + 0.232i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 + 0.556i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.830 + 0.556i$
Analytic conductor: \(9.64580\)
Root analytic conductor: \(3.10576\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1),\ -0.830 + 0.556i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.653613 - 2.14853i\)
\(L(\frac12)\) \(\approx\) \(0.653613 - 2.14853i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + (-2.49 + 1.67i)T \)
59 \( 1 - 7.68iT \)
good5 \( 1 + 7.29iT - 25T^{2} \)
7 \( 1 - 6.28T + 49T^{2} \)
11 \( 1 + 1.79iT - 121T^{2} \)
13 \( 1 - 5.41T + 169T^{2} \)
17 \( 1 - 3.95iT - 289T^{2} \)
19 \( 1 + 0.362T + 361T^{2} \)
23 \( 1 - 18.6iT - 529T^{2} \)
29 \( 1 - 9.98iT - 841T^{2} \)
31 \( 1 + 58.1T + 961T^{2} \)
37 \( 1 - 37.4T + 1.36e3T^{2} \)
41 \( 1 - 50.9iT - 1.68e3T^{2} \)
43 \( 1 - 40.4T + 1.84e3T^{2} \)
47 \( 1 + 49.3iT - 2.20e3T^{2} \)
53 \( 1 + 1.67iT - 2.80e3T^{2} \)
61 \( 1 - 32.2T + 3.72e3T^{2} \)
67 \( 1 + 5.04T + 4.48e3T^{2} \)
71 \( 1 - 68.9iT - 5.04e3T^{2} \)
73 \( 1 - 48.4T + 5.32e3T^{2} \)
79 \( 1 + 61.0T + 6.24e3T^{2} \)
83 \( 1 + 138. iT - 6.88e3T^{2} \)
89 \( 1 - 121. iT - 7.92e3T^{2} \)
97 \( 1 - 115.T + 9.40e3T^{2} \)
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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

−11.10758257139267987582392173342, −9.704669593963043300337511026443, −8.918103690775122460491352063678, −8.335535076131757717631118325111, −7.48153996487182502081599689132, −5.75429772311338050251053682841, −4.65264858936605080288756173933, −3.59914698390517540613345390115, −1.93294625205452365147901640934, −1.03290557018715487261510027021, 2.24405883494962738216001706927, 3.51979711280040077334697364059, 4.58435792300610581807728803761, 5.88113453271207515105616832666, 7.12453245347490538236424817635, 7.74533065657362316885745665591, 8.727746526251464682149525548646, 9.687409423006612822504568506759, 10.71974772044660615376013623215, 11.18478995483163312152760966900

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