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Properties

Label 2-354-177.176-c5-0-47
Degree $2$
Conductor $354$
Sign $0.466 + 0.884i$
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 + (−3.54 − 15.1i)3-s + 16·4-s − 29.8i·5-s + (14.1 + 60.7i)6-s + 197.·7-s − 64·8-s + (−217. + 107. i)9-s + 119. i·10-s − 108.·11-s + (−56.6 − 242. i)12-s + 112. i·13-s − 791.·14-s + (−453. + 105. i)15-s + 256·16-s − 732. i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.227 − 0.973i)3-s + 0.5·4-s − 0.534i·5-s + (0.160 + 0.688i)6-s + 1.52·7-s − 0.353·8-s + (−0.896 + 0.442i)9-s + 0.377i·10-s − 0.270·11-s + (−0.113 − 0.486i)12-s + 0.184i·13-s − 1.07·14-s + (−0.520 + 0.121i)15-s + 0.250·16-s − 0.614i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.466 + 0.884i$
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: \((2,\ 354,\ (\ :5/2),\ 0.466 + 0.884i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.735023749\)
\(L(\frac12)\) \(\approx\) \(1.735023749\)
\(L(\frac{7}{2})\) 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 \( 1 + (3.54 + 15.1i)T \)
59 \( 1 + (-2.58e4 + 6.76e3i)T \)
good5 \( 1 + 29.8iT - 3.12e3T^{2} \)
7 \( 1 - 197.T + 1.68e4T^{2} \)
11 \( 1 + 108.T + 1.61e5T^{2} \)
13 \( 1 - 112. iT - 3.71e5T^{2} \)
17 \( 1 + 732. iT - 1.41e6T^{2} \)
19 \( 1 - 3.04e3T + 2.47e6T^{2} \)
23 \( 1 + 135.T + 6.43e6T^{2} \)
29 \( 1 - 1.91e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.87e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.35e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.51e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.94e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.29e4T + 2.29e8T^{2} \)
53 \( 1 + 1.65e3iT - 4.18e8T^{2} \)
61 \( 1 - 5.26e3iT - 8.44e8T^{2} \)
67 \( 1 - 5.91e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.68e4iT - 1.80e9T^{2} \)
73 \( 1 - 1.75e3iT - 2.07e9T^{2} \)
79 \( 1 - 2.01e4T + 3.07e9T^{2} \)
83 \( 1 + 8.28e4T + 3.93e9T^{2} \)
89 \( 1 - 6.47e4T + 5.58e9T^{2} \)
97 \( 1 - 2.50e4iT - 8.58e9T^{2} \)
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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

−10.68763876942000838194612922411, −9.306606490008994908439880334339, −8.476050864679403362189089318888, −7.68311391891891395239372923539, −7.05800015755371144319162468765, −5.56939428326657995498707365891, −4.86931265678212391201260501833, −2.81815773709111487347530532734, −1.49491681207641952709179258102, −0.876400596821983796548767317905, 0.858975873380137932180804487087, 2.38130756979085301791014977500, 3.69279452037779278586412100875, 4.98758707098979174740770559396, 5.80315751540582803728514466621, 7.26903194471814997505162251761, 8.112504474985102180458689116876, 8.986583007862335597254195114040, 10.05187018185015601230871900348, 10.67269936518745166519198384735

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