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Properties

Label 2-840-35.27-c1-0-21
Degree $2$
Conductor $840$
Sign $-0.861 + 0.508i$
Analytic cond. $6.70743$
Root an. cond. $2.58987$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)3-s + (−2.17 + 0.513i)5-s + (0.659 − 2.56i)7-s + 1.00i·9-s − 1.57·11-s + (−3.14 − 3.14i)13-s + (−1.90 − 1.17i)15-s + (−4.47 + 4.47i)17-s − 6.38·19-s + (2.27 − 1.34i)21-s + (1.38 − 1.38i)23-s + (4.47 − 2.23i)25-s + (−0.707 + 0.707i)27-s + 2.19i·29-s + 1.53i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.973 + 0.229i)5-s + (0.249 − 0.968i)7-s + 0.333i·9-s − 0.475·11-s + (−0.872 − 0.872i)13-s + (−0.491 − 0.303i)15-s + (−1.08 + 1.08i)17-s − 1.46·19-s + (0.497 − 0.293i)21-s + (0.288 − 0.288i)23-s + (0.894 − 0.446i)25-s + (−0.136 + 0.136i)27-s + 0.407i·29-s + 0.276i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 + 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.861 + 0.508i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.861 + 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0641914 - 0.234980i\)
\(L(\frac12)\) \(\approx\) \(0.0641914 - 0.234980i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (2.17 - 0.513i)T \)
7 \( 1 + (-0.659 + 2.56i)T \)
good11 \( 1 + 1.57T + 11T^{2} \)
13 \( 1 + (3.14 + 3.14i)T + 13iT^{2} \)
17 \( 1 + (4.47 - 4.47i)T - 17iT^{2} \)
19 \( 1 + 6.38T + 19T^{2} \)
23 \( 1 + (-1.38 + 1.38i)T - 23iT^{2} \)
29 \( 1 - 2.19iT - 29T^{2} \)
31 \( 1 - 1.53iT - 31T^{2} \)
37 \( 1 + (8.06 + 8.06i)T + 37iT^{2} \)
41 \( 1 + 4.79iT - 41T^{2} \)
43 \( 1 + (-0.0831 + 0.0831i)T - 43iT^{2} \)
47 \( 1 + (3.14 - 3.14i)T - 47iT^{2} \)
53 \( 1 + (-6.30 + 6.30i)T - 53iT^{2} \)
59 \( 1 + 7.59T + 59T^{2} \)
61 \( 1 + 3.73iT - 61T^{2} \)
67 \( 1 + (1.84 + 1.84i)T + 67iT^{2} \)
71 \( 1 - 9.63T + 71T^{2} \)
73 \( 1 + (-2.46 - 2.46i)T + 73iT^{2} \)
79 \( 1 - 15.8iT - 79T^{2} \)
83 \( 1 + (-2.70 - 2.70i)T + 83iT^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + (-8.82 + 8.82i)T - 97iT^{2} \)
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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

−10.12370808119715227100874026867, −8.762429130961808557585326368244, −8.210817101248041408170893506986, −7.38484922526111259511747340077, −6.63059772361034038779369130077, −5.12178869490755379831071163910, −4.26548093472227542424542702667, −3.53865119879142766447767664489, −2.24675355867461167382666982405, −0.10446751945487242842477442160, 2.01943058265364343630471144705, 2.93527579193236690475506130604, 4.37901636810145262975710103576, 5.02011737491040687433906516731, 6.44180027102204482870414342026, 7.19318325490584891694073605039, 8.102310991163450359741347946996, 8.787206008339035397207487712424, 9.397618799824882770926958038426, 10.66038531767169450103977909780

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