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Properties

Label 2-354-177.176-c5-0-84
Degree $2$
Conductor $354$
Sign $0.295 + 0.955i$
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 + (−9.68 + 12.2i)3-s + 16·4-s + 48.6i·5-s + (−38.7 + 48.8i)6-s + 149.·7-s + 64·8-s + (−55.5 − 236. i)9-s + 194. i·10-s − 118.·11-s + (−154. + 195. i)12-s − 964. i·13-s + 596.·14-s + (−593. − 470. i)15-s + 256·16-s − 1.57e3i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.621 + 0.783i)3-s + 0.5·4-s + 0.869i·5-s + (−0.439 + 0.554i)6-s + 1.15·7-s + 0.353·8-s + (−0.228 − 0.973i)9-s + 0.614i·10-s − 0.294·11-s + (−0.310 + 0.391i)12-s − 1.58i·13-s + 0.813·14-s + (−0.681 − 0.540i)15-s + 0.250·16-s − 1.32i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.295 + 0.955i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.295 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.295 + 0.955i$
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: \((2,\ 354,\ (\ :5/2),\ 0.295 + 0.955i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.531865730\)
\(L(\frac12)\) \(\approx\) \(1.531865730\)
\(L(\frac{7}{2})\) 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 \( 1 + (9.68 - 12.2i)T \)
59 \( 1 + (-1.51e4 + 2.20e4i)T \)
good5 \( 1 - 48.6iT - 3.12e3T^{2} \)
7 \( 1 - 149.T + 1.68e4T^{2} \)
11 \( 1 + 118.T + 1.61e5T^{2} \)
13 \( 1 + 964. iT - 3.71e5T^{2} \)
17 \( 1 + 1.57e3iT - 1.41e6T^{2} \)
19 \( 1 + 2.97e3T + 2.47e6T^{2} \)
23 \( 1 + 3.40e3T + 6.43e6T^{2} \)
29 \( 1 - 4.80e3iT - 2.05e7T^{2} \)
31 \( 1 + 6.75e3iT - 2.86e7T^{2} \)
37 \( 1 + 5.99e3iT - 6.93e7T^{2} \)
41 \( 1 + 7.42e3iT - 1.15e8T^{2} \)
43 \( 1 + 6.93e3iT - 1.47e8T^{2} \)
47 \( 1 + 2.61e4T + 2.29e8T^{2} \)
53 \( 1 - 7.64e3iT - 4.18e8T^{2} \)
61 \( 1 - 1.22e4iT - 8.44e8T^{2} \)
67 \( 1 + 3.83e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.43e4iT - 1.80e9T^{2} \)
73 \( 1 + 5.02e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.84e4T + 3.07e9T^{2} \)
83 \( 1 - 1.62e4T + 3.93e9T^{2} \)
89 \( 1 + 2.63e4T + 5.58e9T^{2} \)
97 \( 1 - 4.29e4iT - 8.58e9T^{2} \)
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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

−10.75104452938122075172396382186, −9.979582358726789425119467258620, −8.480174540021743282315920279243, −7.47295695040666092446367902275, −6.31723750311047644720445165907, −5.39388878004260008052578878198, −4.59538004510284699219115611884, −3.44794267148902708080198328683, −2.27608930673861994707323639332, −0.30559120401881599475740575687, 1.49672437754912441262559851140, 2.02759057614549716066813867667, 4.29585687617359709243019747205, 4.77023210695120274873313166802, 6.00799594912428986991150454010, 6.71716845287654624456874429992, 8.142378330577940553025956741862, 8.478625113668311045388952310688, 10.23192945991751557129896953390, 11.21529248427532013812272102711

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