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Properties

Label 2-354-177.176-c5-0-42
Degree $2$
Conductor $354$
Sign $0.432 - 0.901i$
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 + (−15.5 − 1.23i)3-s + 16·4-s + 14.2i·5-s + (62.1 + 4.95i)6-s + 129.·7-s − 64·8-s + (239. + 38.5i)9-s − 56.8i·10-s + 605.·11-s + (−248. − 19.8i)12-s + 1.14e3i·13-s − 516.·14-s + (17.6 − 220. i)15-s + 256·16-s + 1.59e3i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.996 − 0.0795i)3-s + 0.5·4-s + 0.254i·5-s + (0.704 + 0.0562i)6-s + 0.995·7-s − 0.353·8-s + (0.987 + 0.158i)9-s − 0.179i·10-s + 1.50·11-s + (−0.498 − 0.0397i)12-s + 1.87i·13-s − 0.703·14-s + (0.0202 − 0.253i)15-s + 0.250·16-s + 1.33i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.432 - 0.901i$
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: \((2,\ 354,\ (\ :5/2),\ 0.432 - 0.901i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.460130605\)
\(L(\frac12)\) \(\approx\) \(1.460130605\)
\(L(\frac{7}{2})\) 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 \( 1 + (15.5 + 1.23i)T \)
59 \( 1 + (-9.61e3 + 2.49e4i)T \)
good5 \( 1 - 14.2iT - 3.12e3T^{2} \)
7 \( 1 - 129.T + 1.68e4T^{2} \)
11 \( 1 - 605.T + 1.61e5T^{2} \)
13 \( 1 - 1.14e3iT - 3.71e5T^{2} \)
17 \( 1 - 1.59e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.35e3T + 2.47e6T^{2} \)
23 \( 1 - 2.75e3T + 6.43e6T^{2} \)
29 \( 1 + 4.29e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.10e3iT - 2.86e7T^{2} \)
37 \( 1 - 6.12e3iT - 6.93e7T^{2} \)
41 \( 1 - 8.51e3iT - 1.15e8T^{2} \)
43 \( 1 - 783. iT - 1.47e8T^{2} \)
47 \( 1 + 1.63e4T + 2.29e8T^{2} \)
53 \( 1 + 3.29e4iT - 4.18e8T^{2} \)
61 \( 1 - 4.46e4iT - 8.44e8T^{2} \)
67 \( 1 - 6.51e3iT - 1.35e9T^{2} \)
71 \( 1 + 7.03e4iT - 1.80e9T^{2} \)
73 \( 1 - 3.78e4iT - 2.07e9T^{2} \)
79 \( 1 - 8.89e4T + 3.07e9T^{2} \)
83 \( 1 - 3.23e4T + 3.93e9T^{2} \)
89 \( 1 + 6.34e4T + 5.58e9T^{2} \)
97 \( 1 - 1.74e5iT - 8.58e9T^{2} \)
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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

−11.09629686984293402649575447509, −9.860829181128884079415109132781, −9.082179594431415212840862592803, −7.984185001918051383823105932214, −6.76770301329435657738267486051, −6.40212896424063712905759155290, −4.91231002130807893896653953446, −3.92306261097973904760354829475, −1.79861938786661194463925434050, −1.13082161974638800456441153491, 0.72414226003624139327055111691, 1.29113291456005556724183020146, 3.21413451958321888944720557485, 4.85408266262859004432481774798, 5.48323367635579367899309295627, 6.83377717045352427787122220656, 7.53631484526985585467217659767, 8.746225590863430201790779538776, 9.549829998868812403854091872795, 10.66510616187835085222454583943

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