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Properties

Label 2-35-35.23-c2-0-1
Degree $2$
Conductor $35$
Sign $0.898 - 0.438i$
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  + (−0.156 − 0.585i)2-s + (−1.07 + 4.00i)3-s + (3.14 − 1.81i)4-s + (4.12 + 2.83i)5-s + 2.51·6-s + (−5.97 − 3.65i)7-s + (−3.27 − 3.27i)8-s + (−7.05 − 4.07i)9-s + (1.01 − 2.85i)10-s + (−2.32 − 4.01i)11-s + (3.89 + 14.5i)12-s + (−6.67 − 6.67i)13-s + (−1.20 + 4.07i)14-s + (−15.7 + 13.4i)15-s + (5.86 − 10.1i)16-s + (18.4 + 4.93i)17-s + ⋯
L(s)  = 1  + (−0.0784 − 0.292i)2-s + (−0.357 + 1.33i)3-s + (0.786 − 0.454i)4-s + (0.824 + 0.566i)5-s + 0.418·6-s + (−0.853 − 0.521i)7-s + (−0.408 − 0.408i)8-s + (−0.784 − 0.452i)9-s + (0.101 − 0.285i)10-s + (−0.210 − 0.365i)11-s + (0.324 + 1.21i)12-s + (−0.513 − 0.513i)13-s + (−0.0857 + 0.290i)14-s + (−1.04 + 0.896i)15-s + (0.366 − 0.634i)16-s + (1.08 + 0.290i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.898 - 0.438i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ 0.898 - 0.438i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.02249 + 0.235914i\)
\(L(\frac12)\) \(\approx\) \(1.02249 + 0.235914i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-4.12 - 2.83i)T \)
7 \( 1 + (5.97 + 3.65i)T \)
good2 \( 1 + (0.156 + 0.585i)T + (-3.46 + 2i)T^{2} \)
3 \( 1 + (1.07 - 4.00i)T + (-7.79 - 4.5i)T^{2} \)
11 \( 1 + (2.32 + 4.01i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (6.67 + 6.67i)T + 169iT^{2} \)
17 \( 1 + (-18.4 - 4.93i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (22.5 + 13.0i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-7.71 + 2.06i)T + (458. - 264.5i)T^{2} \)
29 \( 1 - 39.4iT - 841T^{2} \)
31 \( 1 + (-14.1 - 24.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (9.10 + 33.9i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 20.8T + 1.68e3T^{2} \)
43 \( 1 + (14.8 + 14.8i)T + 1.84e3iT^{2} \)
47 \( 1 + (-15.3 - 57.3i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-8.55 + 31.9i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (36.4 - 21.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-19.6 + 33.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-35.2 - 9.45i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 41.3T + 5.04e3T^{2} \)
73 \( 1 + (-1.80 + 6.72i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-119. - 68.8i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (37.7 + 37.7i)T + 6.88e3iT^{2} \)
89 \( 1 + (106. + 61.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (4.90 - 4.90i)T - 9.40e3iT^{2} \)
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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

−16.35653843358201242564494546999, −15.35245080762193187810556470565, −14.37408485534366523919826858815, −12.67675176870642807101614486224, −10.79315103553657117353246710992, −10.43836576461481391849300683384, −9.468522699828861632231687852346, −6.79607245668221255302363084093, −5.45333531226109982364118687759, −3.16979462412432104339975255657, 2.20901876764480877450131633870, 5.87755336785752030409153789083, 6.77917246698702942851428666228, 8.154291523549803186750970461559, 9.862987599723089520569652422464, 11.93752277533134476408702587844, 12.49644156836369152007098302246, 13.50239690243749363107655325810, 15.17941648781262902249706679663, 16.70538076131332416949288071656

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