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Properties

Label 2-354-177.176-c1-0-10
Degree $2$
Conductor $354$
Sign $-0.652 + 0.757i$
Analytic cond. $2.82670$
Root an. cond. $1.68128$
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  − 2-s + (−1.65 − 0.509i)3-s + 4-s + 2.90i·5-s + (1.65 + 0.509i)6-s − 3.06·7-s − 8-s + (2.48 + 1.68i)9-s − 2.90i·10-s + 1.35·11-s + (−1.65 − 0.509i)12-s − 7.06i·13-s + 3.06·14-s + (1.48 − 4.80i)15-s + 16-s + 0.568i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.955 − 0.294i)3-s + 0.5·4-s + 1.29i·5-s + (0.675 + 0.208i)6-s − 1.16·7-s − 0.353·8-s + (0.826 + 0.562i)9-s − 0.917i·10-s + 0.408·11-s + (−0.477 − 0.147i)12-s − 1.95i·13-s + 0.820·14-s + (0.382 − 1.24i)15-s + 0.250·16-s + 0.137i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.652 + 0.757i$
Analytic conductor: \(2.82670\)
Root analytic conductor: \(1.68128\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1/2),\ -0.652 + 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0771695 - 0.168225i\)
\(L(\frac12)\) \(\approx\) \(0.0771695 - 0.168225i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (1.65 + 0.509i)T \)
59 \( 1 + (3.07 - 7.03i)T \)
good5 \( 1 - 2.90iT - 5T^{2} \)
7 \( 1 + 3.06T + 7T^{2} \)
11 \( 1 - 1.35T + 11T^{2} \)
13 \( 1 + 7.06iT - 13T^{2} \)
17 \( 1 - 0.568iT - 17T^{2} \)
19 \( 1 + 4.42T + 19T^{2} \)
23 \( 1 + 7.20T + 23T^{2} \)
29 \( 1 + 2.41iT - 29T^{2} \)
31 \( 1 + 6.44iT - 31T^{2} \)
37 \( 1 + 1.74iT - 37T^{2} \)
41 \( 1 + 8.09iT - 41T^{2} \)
43 \( 1 + 0.770iT - 43T^{2} \)
47 \( 1 + 4.48T + 47T^{2} \)
53 \( 1 + 1.03iT - 53T^{2} \)
61 \( 1 - 10.6iT - 61T^{2} \)
67 \( 1 + 1.09iT - 67T^{2} \)
71 \( 1 + 11.0iT - 71T^{2} \)
73 \( 1 - 5.96iT - 73T^{2} \)
79 \( 1 - 8.62T + 79T^{2} \)
83 \( 1 - 14.6T + 83T^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 + 8.16iT - 97T^{2} \)
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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

−10.77949573523982990776007732215, −10.39800936302582037568586425769, −9.679437953916926383254515249284, −8.095382421159393350405615997642, −7.23902417612831251061127717250, −6.29519409275639894133831699625, −5.84088148432443818301566409165, −3.75261485573593655758622436758, −2.45531379445463472890451824723, −0.17678319186235787895191156627, 1.58219246825616697157073117853, 3.90927441675529685744337903036, 4.87720500921841065151384919377, 6.33070967848373999932579008822, 6.70378028772140510325561062858, 8.332230590295464417747235007464, 9.361199892468444654991654407063, 9.639367095281897060055677130481, 10.83749523356389602737419275308, 11.89476800439326094058070956356

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