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Properties

Label 2-35-5.2-c2-0-5
Degree $2$
Conductor $35$
Sign $-0.707 + 0.706i$
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  + (−2.47 − 2.47i)2-s + (2.98 − 2.98i)3-s + 8.24i·4-s + (−4.25 − 2.63i)5-s − 14.7·6-s + (1.87 + 1.87i)7-s + (10.5 − 10.5i)8-s − 8.76i·9-s + (4.00 + 17.0i)10-s + 11.1·11-s + (24.5 + 24.5i)12-s + (7.26 − 7.26i)13-s − 9.25i·14-s + (−20.5 + 4.81i)15-s − 19.0·16-s + (−2.00 − 2.00i)17-s + ⋯
L(s)  = 1  + (−1.23 − 1.23i)2-s + (0.993 − 0.993i)3-s + 2.06i·4-s + (−0.850 − 0.526i)5-s − 2.45·6-s + (0.267 + 0.267i)7-s + (1.31 − 1.31i)8-s − 0.973i·9-s + (0.400 + 1.70i)10-s + 1.01·11-s + (2.04 + 2.04i)12-s + (0.558 − 0.558i)13-s − 0.661i·14-s + (−1.36 + 0.321i)15-s − 1.19·16-s + (−0.117 − 0.117i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.706i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.707 + 0.706i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ -0.707 + 0.706i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.272303 - 0.658451i\)
\(L(\frac12)\) \(\approx\) \(0.272303 - 0.658451i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (4.25 + 2.63i)T \)
7 \( 1 + (-1.87 - 1.87i)T \)
good2 \( 1 + (2.47 + 2.47i)T + 4iT^{2} \)
3 \( 1 + (-2.98 + 2.98i)T - 9iT^{2} \)
11 \( 1 - 11.1T + 121T^{2} \)
13 \( 1 + (-7.26 + 7.26i)T - 169iT^{2} \)
17 \( 1 + (2.00 + 2.00i)T + 289iT^{2} \)
19 \( 1 - 5.59iT - 361T^{2} \)
23 \( 1 + (6.88 - 6.88i)T - 529iT^{2} \)
29 \( 1 - 46.4iT - 841T^{2} \)
31 \( 1 + 23.7T + 961T^{2} \)
37 \( 1 + (14.8 + 14.8i)T + 1.36e3iT^{2} \)
41 \( 1 - 31.9T + 1.68e3T^{2} \)
43 \( 1 + (3.79 - 3.79i)T - 1.84e3iT^{2} \)
47 \( 1 + (20.1 + 20.1i)T + 2.20e3iT^{2} \)
53 \( 1 + (73.4 - 73.4i)T - 2.80e3iT^{2} \)
59 \( 1 + 54.9iT - 3.48e3T^{2} \)
61 \( 1 - 96.1T + 3.72e3T^{2} \)
67 \( 1 + (2.50 + 2.50i)T + 4.48e3iT^{2} \)
71 \( 1 + 104.T + 5.04e3T^{2} \)
73 \( 1 + (-20.1 + 20.1i)T - 5.32e3iT^{2} \)
79 \( 1 + 100. iT - 6.24e3T^{2} \)
83 \( 1 + (8.15 - 8.15i)T - 6.88e3iT^{2} \)
89 \( 1 + 39.5iT - 7.92e3T^{2} \)
97 \( 1 + (52.3 + 52.3i)T + 9.40e3iT^{2} \)
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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

−16.23148037577188160591348804267, −14.50808754446144130776875719751, −12.92648590143774285846048120780, −12.15461294544348550402473442080, −11.02319688270383691137347889750, −9.148998154252284382158887635694, −8.444055512432407676646298873380, −7.43401781260154160857717604822, −3.42972495969884600531389038533, −1.46537882802061874436703104981, 4.06862187102391280575202432718, 6.61638849786222016869946348715, 7.992990188808806597928827463569, 8.918848257002547574882636489727, 9.973189305403015079993780646582, 11.30314935192048830297897640770, 14.15039201118808811301232787599, 14.81510410715023727579082250173, 15.68570801231674348653601855130, 16.47541799230392717994319973930

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