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Properties

Label 2-354-177.176-c5-0-87
Degree $2$
Conductor $354$
Sign $-0.914 + 0.405i$
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 + (9.44 − 12.4i)3-s + 16·4-s − 32.9i·5-s + (−37.7 + 49.6i)6-s + 193.·7-s − 64·8-s + (−64.6 − 234. i)9-s + 131. i·10-s − 33.0·11-s + (151. − 198. i)12-s + 210. i·13-s − 773.·14-s + (−409. − 311. i)15-s + 256·16-s − 1.12e3i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.605 − 0.795i)3-s + 0.5·4-s − 0.590i·5-s + (−0.428 + 0.562i)6-s + 1.49·7-s − 0.353·8-s + (−0.266 − 0.963i)9-s + 0.417i·10-s − 0.0824·11-s + (0.302 − 0.397i)12-s + 0.346i·13-s − 1.05·14-s + (−0.469 − 0.357i)15-s + 0.250·16-s − 0.941i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.405i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.914 + 0.405i$
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: \((2,\ 354,\ (\ :5/2),\ -0.914 + 0.405i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.628125242\)
\(L(\frac12)\) \(\approx\) \(1.628125242\)
\(L(\frac{7}{2})\) 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 \( 1 + (-9.44 + 12.4i)T \)
59 \( 1 + (-2.34e4 - 1.28e4i)T \)
good5 \( 1 + 32.9iT - 3.12e3T^{2} \)
7 \( 1 - 193.T + 1.68e4T^{2} \)
11 \( 1 + 33.0T + 1.61e5T^{2} \)
13 \( 1 - 210. iT - 3.71e5T^{2} \)
17 \( 1 + 1.12e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.67e3T + 2.47e6T^{2} \)
23 \( 1 + 2.56e3T + 6.43e6T^{2} \)
29 \( 1 - 1.32e3iT - 2.05e7T^{2} \)
31 \( 1 + 5.85e3iT - 2.86e7T^{2} \)
37 \( 1 - 3.79e3iT - 6.93e7T^{2} \)
41 \( 1 + 4.35e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.69e4iT - 1.47e8T^{2} \)
47 \( 1 + 3.28e3T + 2.29e8T^{2} \)
53 \( 1 + 2.82e4iT - 4.18e8T^{2} \)
61 \( 1 - 9.80e3iT - 8.44e8T^{2} \)
67 \( 1 - 3.40e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.24e4iT - 1.80e9T^{2} \)
73 \( 1 - 2.20e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.42e4T + 3.07e9T^{2} \)
83 \( 1 - 9.68e4T + 3.93e9T^{2} \)
89 \( 1 + 1.13e5T + 5.58e9T^{2} \)
97 \( 1 + 1.82e5iT - 8.58e9T^{2} \)
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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

−10.10157494082139597570175420823, −8.923467372621657502332121556386, −8.431064977651504788954522220051, −7.66286920762184089582334055767, −6.74333554348018441089423554243, −5.40425419427632990807528981011, −4.15118595590136723468613033796, −2.39786370542895646584063654242, −1.59159984321821738891763882797, −0.46104244835350934405430851662, 1.58720416401804565985290047729, 2.61048501743026843619742893389, 3.96377500880656880496149711319, 5.03587885854271449546084115190, 6.34634591440974545327371154809, 7.78572445310350907447933247694, 8.214045097671578782135812469025, 9.087515669716335893955281649705, 10.34946278100724343367275553262, 10.69010461254650163888266759266

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