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Properties

Label 2-354-177.176-c5-0-62
Degree $2$
Conductor $354$
Sign $-0.961 + 0.276i$
Analytic cond. $56.7758$
Root an. cond. $7.53497$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + (−4.24 − 15.0i)3-s + 16·4-s + 2.81i·5-s + (16.9 + 60.0i)6-s − 187.·7-s − 64·8-s + (−207. + 127. i)9-s − 11.2i·10-s + 666.·11-s + (−67.8 − 240. i)12-s + 37.6i·13-s + 748.·14-s + (42.2 − 11.9i)15-s + 256·16-s + 547. i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.272 − 0.962i)3-s + 0.5·4-s + 0.0503i·5-s + (0.192 + 0.680i)6-s − 1.44·7-s − 0.353·8-s + (−0.852 + 0.523i)9-s − 0.0355i·10-s + 1.66·11-s + (−0.136 − 0.481i)12-s + 0.0617i·13-s + 1.02·14-s + (0.0484 − 0.0136i)15-s + 0.250·16-s + 0.459i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.276i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.961 + 0.276i$
Analytic conductor: \(56.7758\)
Root analytic conductor: \(7.53497\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :5/2),\ -0.961 + 0.276i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.5963064289\)
\(L(\frac12)\) \(\approx\) \(0.5963064289\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4T \)
3 \( 1 + (4.24 + 15.0i)T \)
59 \( 1 + (-113. - 2.67e4i)T \)
good5 \( 1 - 2.81iT - 3.12e3T^{2} \)
7 \( 1 + 187.T + 1.68e4T^{2} \)
11 \( 1 - 666.T + 1.61e5T^{2} \)
13 \( 1 - 37.6iT - 3.71e5T^{2} \)
17 \( 1 - 547. iT - 1.41e6T^{2} \)
19 \( 1 - 2.12e3T + 2.47e6T^{2} \)
23 \( 1 + 3.66e3T + 6.43e6T^{2} \)
29 \( 1 + 7.84e3iT - 2.05e7T^{2} \)
31 \( 1 + 9.40e3iT - 2.86e7T^{2} \)
37 \( 1 - 2.00e3iT - 6.93e7T^{2} \)
41 \( 1 - 6.35e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.09e4iT - 1.47e8T^{2} \)
47 \( 1 - 5.89e3T + 2.29e8T^{2} \)
53 \( 1 + 1.05e4iT - 4.18e8T^{2} \)
61 \( 1 + 3.75e4iT - 8.44e8T^{2} \)
67 \( 1 + 1.08e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.40e4iT - 1.80e9T^{2} \)
73 \( 1 - 5.46e3iT - 2.07e9T^{2} \)
79 \( 1 + 3.36e4T + 3.07e9T^{2} \)
83 \( 1 - 1.38e4T + 3.93e9T^{2} \)
89 \( 1 + 6.65e4T + 5.58e9T^{2} \)
97 \( 1 + 1.45e5iT - 8.58e9T^{2} \)
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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

−9.900093208224659865659987939560, −9.453716373792592705788706743291, −8.286969108293412431323877085795, −7.31565866260624304100566042265, −6.36707397725520775496753250223, −5.99648827567799148309141217849, −3.89797627656725332201183903993, −2.64654075127959165002280928262, −1.29638049810194185905607826962, −0.24647552572206402724687899612, 1.08259304087679050749366935723, 3.07246620389445395973510102523, 3.78538379723452882272290905938, 5.30764317373758672942223499554, 6.41478386284104209491224845189, 7.09448889656470451755361962059, 8.832065721497209723229145194155, 9.214136758321016292921188126628, 10.01696992006689944175393379805, 10.76044586538655920522955137978

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