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Properties

Label 2-354-3.2-c2-0-5
Degree $2$
Conductor $354$
Sign $-0.985 + 0.168i$
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 + (2.95 − 0.505i)3-s − 2.00·4-s + 6.09i·5-s + (0.714 + 4.18i)6-s − 12.9·7-s − 2.82i·8-s + (8.48 − 2.98i)9-s − 8.61·10-s + 9.79i·11-s + (−5.91 + 1.01i)12-s − 24.3·13-s − 18.2i·14-s + (3.07 + 18.0i)15-s + 4.00·16-s − 24.3i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.985 − 0.168i)3-s − 0.500·4-s + 1.21i·5-s + (0.119 + 0.697i)6-s − 1.84·7-s − 0.353i·8-s + (0.943 − 0.332i)9-s − 0.861·10-s + 0.890i·11-s + (−0.492 + 0.0842i)12-s − 1.87·13-s − 1.30i·14-s + (0.205 + 1.20i)15-s + 0.250·16-s − 1.43i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.168i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.985 + 0.168i$
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: \((2,\ 354,\ (\ :1),\ -0.985 + 0.168i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0783349 - 0.923650i\)
\(L(\frac12)\) \(\approx\) \(0.0783349 - 0.923650i\)
\(L(2)\) 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 \( 1 + (-2.95 + 0.505i)T \)
59 \( 1 + 7.68iT \)
good5 \( 1 - 6.09iT - 25T^{2} \)
7 \( 1 + 12.9T + 49T^{2} \)
11 \( 1 - 9.79iT - 121T^{2} \)
13 \( 1 + 24.3T + 169T^{2} \)
17 \( 1 + 24.3iT - 289T^{2} \)
19 \( 1 - 0.781T + 361T^{2} \)
23 \( 1 - 14.4iT - 529T^{2} \)
29 \( 1 - 32.9iT - 841T^{2} \)
31 \( 1 - 1.57T + 961T^{2} \)
37 \( 1 + 27.3T + 1.36e3T^{2} \)
41 \( 1 - 72.7iT - 1.68e3T^{2} \)
43 \( 1 + 2.22T + 1.84e3T^{2} \)
47 \( 1 - 43.9iT - 2.20e3T^{2} \)
53 \( 1 - 40.0iT - 2.80e3T^{2} \)
61 \( 1 - 43.8T + 3.72e3T^{2} \)
67 \( 1 + 40.6T + 4.48e3T^{2} \)
71 \( 1 + 96.3iT - 5.04e3T^{2} \)
73 \( 1 + 122.T + 5.32e3T^{2} \)
79 \( 1 - 18.6T + 6.24e3T^{2} \)
83 \( 1 + 42.6iT - 6.88e3T^{2} \)
89 \( 1 - 108. iT - 7.92e3T^{2} \)
97 \( 1 - 21.4T + 9.40e3T^{2} \)
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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

−12.06104268501043131698344425428, −10.28302793173293958364620273992, −9.684009626707779867147447180368, −9.220610155791819121070864079749, −7.39453535094941408578844072354, −7.23110407472241587084719245724, −6.43509316161377796970234157722, −4.78594528718350565964723858402, −3.25767157049412412665594623416, −2.65567891791344484042867333362, 0.34958842330388870535433289122, 2.26634842763659969809112205934, 3.40497314098080727794180178871, 4.34732760475150583829830004980, 5.66945812415012268060720878966, 7.08362635427803429826294338889, 8.423275546380175504811090679030, 8.966722729011883903964734774094, 9.898274152229883754361984295829, 10.29424292210146315603428929570

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