Properties

Label 2-35-35.34-c4-0-7
Degree $2$
Conductor $35$
Sign $0.828 + 0.559i$
Analytic cond. $3.61794$
Root an. cond. $1.90209$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.44i·2-s + 5·3-s + 10·4-s + (5 + 24.4i)5-s − 12.2i·6-s + (35 − 34.2i)7-s − 63.6i·8-s − 56·9-s + (59.9 − 12.2i)10-s + 89·11-s + 50·12-s − 5·13-s + (−84 − 85.7i)14-s + (25 + 122. i)15-s + 4.00·16-s − 485·17-s + ⋯
L(s)  = 1  − 0.612i·2-s + 0.555·3-s + 0.625·4-s + (0.200 + 0.979i)5-s − 0.340i·6-s + (0.714 − 0.699i)7-s − 0.995i·8-s − 0.691·9-s + (0.599 − 0.122i)10-s + 0.735·11-s + 0.347·12-s − 0.0295·13-s + (−0.428 − 0.437i)14-s + (0.111 + 0.544i)15-s + 0.0156·16-s − 1.67·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 + 0.559i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.828 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.828 + 0.559i$
Analytic conductor: \(3.61794\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :2),\ 0.828 + 0.559i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.88436 - 0.576967i\)
\(L(\frac12)\) \(\approx\) \(1.88436 - 0.576967i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-5 - 24.4i)T \)
7 \( 1 + (-35 + 34.2i)T \)
good2 \( 1 + 2.44iT - 16T^{2} \)
3 \( 1 - 5T + 81T^{2} \)
11 \( 1 - 89T + 1.46e4T^{2} \)
13 \( 1 + 5T + 2.85e4T^{2} \)
17 \( 1 + 485T + 8.35e4T^{2} \)
19 \( 1 - 220. iT - 1.30e5T^{2} \)
23 \( 1 - 700. iT - 2.79e5T^{2} \)
29 \( 1 - 191T + 7.07e5T^{2} \)
31 \( 1 + 1.05e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.63e3iT - 1.87e6T^{2} \)
41 \( 1 - 2.91e3iT - 2.82e6T^{2} \)
43 \( 1 + 377. iT - 3.41e6T^{2} \)
47 \( 1 + 2.19e3T + 4.87e6T^{2} \)
53 \( 1 - 1.58e3iT - 7.89e6T^{2} \)
59 \( 1 + 3.62e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.93e3iT - 1.38e7T^{2} \)
67 \( 1 + 2.04e3iT - 2.01e7T^{2} \)
71 \( 1 - 4.45e3T + 2.54e7T^{2} \)
73 \( 1 - 8.65e3T + 2.83e7T^{2} \)
79 \( 1 - 5.56e3T + 3.89e7T^{2} \)
83 \( 1 - 1.99e3T + 4.74e7T^{2} \)
89 \( 1 + 808. iT - 6.27e7T^{2} \)
97 \( 1 - 9.23e3T + 8.85e7T^{2} \)
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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

−15.36150055342789548375809918933, −14.45065008951463534921524175665, −13.42837995572760950408359374228, −11.51230780852540892505119352806, −11.00590441988255020458470577646, −9.558580304904769233577668359034, −7.73872135544898343163271737015, −6.41831416280636474931610874263, −3.67335978508331130762491268181, −2.07092370770431135743982480020, 2.21157011314326723139645887774, 4.96861067944583778768211865705, 6.49968452293574160235912884544, 8.372013421659663015342318746996, 8.905162000563469516246488216410, 11.15109951002768807288444074118, 12.20360554861326034382482696112, 13.82124915359663082856912551027, 14.85273661187535033468194370028, 15.78605964050010973043571338674

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