Properties

Label 2-354-1.1-c5-0-38
Degree $2$
Conductor $354$
Sign $-1$
Analytic cond. $56.7758$
Root an. cond. $7.53497$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 9·3-s + 16·4-s − 3.84·5-s − 36·6-s + 56.0·7-s − 64·8-s + 81·9-s + 15.3·10-s + 11.9·11-s + 144·12-s − 332.·13-s − 224.·14-s − 34.6·15-s + 256·16-s − 542.·17-s − 324·18-s − 1.03e3·19-s − 61.5·20-s + 504.·21-s − 47.7·22-s + 2.80e3·23-s − 576·24-s − 3.11e3·25-s + 1.33e3·26-s + 729·27-s + 896.·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.0688·5-s − 0.408·6-s + 0.432·7-s − 0.353·8-s + 0.333·9-s + 0.0486·10-s + 0.0297·11-s + 0.288·12-s − 0.546·13-s − 0.305·14-s − 0.0397·15-s + 0.250·16-s − 0.454·17-s − 0.235·18-s − 0.657·19-s − 0.0344·20-s + 0.249·21-s − 0.0210·22-s + 1.10·23-s − 0.204·24-s − 0.995·25-s + 0.386·26-s + 0.192·27-s + 0.216·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(56.7758\)
Root analytic conductor: \(7.53497\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 354,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 - 9T \)
59 \( 1 - 3.48e3T \)
good5 \( 1 + 3.84T + 3.12e3T^{2} \)
7 \( 1 - 56.0T + 1.68e4T^{2} \)
11 \( 1 - 11.9T + 1.61e5T^{2} \)
13 \( 1 + 332.T + 3.71e5T^{2} \)
17 \( 1 + 542.T + 1.41e6T^{2} \)
19 \( 1 + 1.03e3T + 2.47e6T^{2} \)
23 \( 1 - 2.80e3T + 6.43e6T^{2} \)
29 \( 1 + 7.31e3T + 2.05e7T^{2} \)
31 \( 1 - 6.97e3T + 2.86e7T^{2} \)
37 \( 1 + 1.43e4T + 6.93e7T^{2} \)
41 \( 1 - 8.93e3T + 1.15e8T^{2} \)
43 \( 1 + 7.69e3T + 1.47e8T^{2} \)
47 \( 1 - 2.13e3T + 2.29e8T^{2} \)
53 \( 1 - 1.28e4T + 4.18e8T^{2} \)
61 \( 1 - 1.57e4T + 8.44e8T^{2} \)
67 \( 1 - 1.87e4T + 1.35e9T^{2} \)
71 \( 1 + 3.35e4T + 1.80e9T^{2} \)
73 \( 1 + 3.93e3T + 2.07e9T^{2} \)
79 \( 1 + 5.81e4T + 3.07e9T^{2} \)
83 \( 1 - 6.20e4T + 3.93e9T^{2} \)
89 \( 1 + 4.97e4T + 5.58e9T^{2} \)
97 \( 1 + 1.37e5T + 8.58e9T^{2} \)
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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

−10.04580415694351614417350847293, −9.165803481455451176153064459553, −8.386515121789716606118508927583, −7.52158290357035042322686014661, −6.64441718396172687071709854885, −5.24563486241719942591110026727, −3.96394634759925839405450330053, −2.60586968759062582125539098854, −1.56068977163600938526275351376, 0, 1.56068977163600938526275351376, 2.60586968759062582125539098854, 3.96394634759925839405450330053, 5.24563486241719942591110026727, 6.64441718396172687071709854885, 7.52158290357035042322686014661, 8.386515121789716606118508927583, 9.165803481455451176153064459553, 10.04580415694351614417350847293

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