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Properties

Label 2-354-177.176-c5-0-3
Degree $2$
Conductor $354$
Sign $-0.951 + 0.306i$
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 + (−13.3 − 8.07i)3-s + 16·4-s + 83.4i·5-s + (−53.3 − 32.3i)6-s − 123.·7-s + 64·8-s + (112. + 215. i)9-s + 333. i·10-s + 49.5·11-s + (−213. − 129. i)12-s + 349. i·13-s − 495.·14-s + (674. − 1.11e3i)15-s + 256·16-s + 992. i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.855 − 0.518i)3-s + 0.5·4-s + 1.49i·5-s + (−0.604 − 0.366i)6-s − 0.955·7-s + 0.353·8-s + (0.463 + 0.886i)9-s + 1.05i·10-s + 0.123·11-s + (−0.427 − 0.259i)12-s + 0.573i·13-s − 0.675·14-s + (0.773 − 1.27i)15-s + 0.250·16-s + 0.833i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.306i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.951 + 0.306i$
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: \((2,\ 354,\ (\ :5/2),\ -0.951 + 0.306i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.4471094036\)
\(L(\frac12)\) \(\approx\) \(0.4471094036\)
\(L(\frac{7}{2})\) 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 \( 1 + (13.3 + 8.07i)T \)
59 \( 1 + (-1.75e4 + 2.01e4i)T \)
good5 \( 1 - 83.4iT - 3.12e3T^{2} \)
7 \( 1 + 123.T + 1.68e4T^{2} \)
11 \( 1 - 49.5T + 1.61e5T^{2} \)
13 \( 1 - 349. iT - 3.71e5T^{2} \)
17 \( 1 - 992. iT - 1.41e6T^{2} \)
19 \( 1 - 2.08e3T + 2.47e6T^{2} \)
23 \( 1 + 3.91e3T + 6.43e6T^{2} \)
29 \( 1 + 1.00e3iT - 2.05e7T^{2} \)
31 \( 1 + 1.15e3iT - 2.86e7T^{2} \)
37 \( 1 - 2.57e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.21e4iT - 1.15e8T^{2} \)
43 \( 1 + 437. iT - 1.47e8T^{2} \)
47 \( 1 + 2.40e4T + 2.29e8T^{2} \)
53 \( 1 - 1.80e4iT - 4.18e8T^{2} \)
61 \( 1 - 2.68e4iT - 8.44e8T^{2} \)
67 \( 1 + 6.93e4iT - 1.35e9T^{2} \)
71 \( 1 - 1.51e4iT - 1.80e9T^{2} \)
73 \( 1 + 1.72e4iT - 2.07e9T^{2} \)
79 \( 1 + 4.15e4T + 3.07e9T^{2} \)
83 \( 1 + 9.51e4T + 3.93e9T^{2} \)
89 \( 1 + 1.21e5T + 5.58e9T^{2} \)
97 \( 1 + 4.92e4iT - 8.58e9T^{2} \)
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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

−11.35145733579687778789521860971, −10.40472533453480956085948093922, −9.769327710228236907373369886754, −7.86468629151612702951982680985, −6.94440647099206488247659041683, −6.37355130986149793467708429905, −5.64014433550603391148898294002, −4.06658008374171285772921490945, −3.03262418957154079022904181055, −1.80957567877550887264020405913, 0.10306509198156629649839150800, 1.20967480772484811651908189116, 3.20866343791363143181043032228, 4.29529142725937830272971757600, 5.21418820599624126809755627217, 5.83600753867379980509061893013, 6.96997939115539534832834252278, 8.289115022707373258828419294245, 9.613727943021059626831348496683, 9.912244046440626067333805282898

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