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Properties

Label 2-35-7.5-c2-0-1
Degree $2$
Conductor $35$
Sign $-0.339 - 0.940i$
Analytic cond. $0.953680$
Root an. cond. $0.976565$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.242 + 0.420i)2-s + (−4.74 + 2.74i)3-s + (1.88 + 3.25i)4-s + (−1.93 − 1.11i)5-s − 2.66i·6-s + (5.87 + 3.80i)7-s − 3.77·8-s + (10.5 − 18.2i)9-s + (0.941 − 0.543i)10-s + (3.29 + 5.71i)11-s + (−17.8 − 10.3i)12-s + 3.29i·13-s + (−3.02 + 1.55i)14-s + 12.2·15-s + (−6.61 + 11.4i)16-s + (9.44 − 5.45i)17-s + ⋯
L(s)  = 1  + (−0.121 + 0.210i)2-s + (−1.58 + 0.913i)3-s + (0.470 + 0.814i)4-s + (−0.387 − 0.223i)5-s − 0.444i·6-s + (0.839 + 0.542i)7-s − 0.471·8-s + (1.17 − 2.02i)9-s + (0.0941 − 0.0543i)10-s + (0.299 + 0.519i)11-s + (−1.48 − 0.859i)12-s + 0.253i·13-s + (−0.216 + 0.110i)14-s + 0.817·15-s + (−0.413 + 0.715i)16-s + (0.555 − 0.320i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.339 - 0.940i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.339 - 0.940i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ -0.339 - 0.940i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.387335 + 0.551676i\)
\(L(\frac12)\) \(\approx\) \(0.387335 + 0.551676i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.93 + 1.11i)T \)
7 \( 1 + (-5.87 - 3.80i)T \)
good2 \( 1 + (0.242 - 0.420i)T + (-2 - 3.46i)T^{2} \)
3 \( 1 + (4.74 - 2.74i)T + (4.5 - 7.79i)T^{2} \)
11 \( 1 + (-3.29 - 5.71i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 3.29iT - 169T^{2} \)
17 \( 1 + (-9.44 + 5.45i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-8.27 - 4.77i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-5.80 + 10.0i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 23.6T + 841T^{2} \)
31 \( 1 + (-9.22 + 5.32i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-31.3 + 54.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 49.7iT - 1.68e3T^{2} \)
43 \( 1 + 7.82T + 1.84e3T^{2} \)
47 \( 1 + (-26.4 - 15.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-31.0 - 53.8i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (39.0 - 22.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-35.0 - 20.2i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (26.3 + 45.7i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 48.2T + 5.04e3T^{2} \)
73 \( 1 + (25.2 - 14.5i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-5.47 + 9.47i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 14.0iT - 6.88e3T^{2} \)
89 \( 1 + (-91.7 - 52.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 118. iT - 9.40e3T^{2} \)
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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

−16.67861225501239443395018560749, −15.85310657883608783582992589348, −14.89194937672836113328309610989, −12.37274160375773009825583786247, −11.80445555959093349416830961008, −10.85457231838492823154309441672, −9.171895998636798373307960120673, −7.37382791034009729184372926491, −5.71763165968427736534948983525, −4.24826913804754478942692652722, 1.14791504190157114832869762290, 5.19776355699938965592974587622, 6.45525235922783849884417180090, 7.67850126393456253120925271329, 10.28745616488919511093045755140, 11.30199987004154626996393174886, 11.77451838839279118559065473407, 13.38561096801880102611129132567, 14.82075137733540900469721050277, 16.27046566100034371512397479655

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