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Properties

Label 2-35-7.5-c2-0-2
Degree $2$
Conductor $35$
Sign $0.799 - 0.600i$
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  + (−0.410 + 0.711i)2-s + (0.507 − 0.292i)3-s + (1.66 + 2.87i)4-s + (1.93 + 1.11i)5-s + 0.481i·6-s + (1.91 − 6.73i)7-s − 6.01·8-s + (−4.32 + 7.49i)9-s + (−1.59 + 0.918i)10-s + (−6.95 − 12.0i)11-s + (1.68 + 0.974i)12-s − 18.3i·13-s + (4.00 + 4.12i)14-s + 1.31·15-s + (−4.17 + 7.23i)16-s + (−8.98 + 5.19i)17-s + ⋯
L(s)  = 1  + (−0.205 + 0.355i)2-s + (0.169 − 0.0976i)3-s + (0.415 + 0.719i)4-s + (0.387 + 0.223i)5-s + 0.0802i·6-s + (0.273 − 0.961i)7-s − 0.752·8-s + (−0.480 + 0.832i)9-s + (−0.159 + 0.0918i)10-s + (−0.632 − 1.09i)11-s + (0.140 + 0.0811i)12-s − 1.41i·13-s + (0.286 + 0.294i)14-s + 0.0873·15-s + (−0.261 + 0.452i)16-s + (−0.528 + 0.305i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.799 - 0.600i$
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: \((2,\ 35,\ (\ :1),\ 0.799 - 0.600i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.998559 + 0.333035i\)
\(L(\frac12)\) \(\approx\) \(0.998559 + 0.333035i\)
\(L(2)\) 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 \( 1 + (-1.91 + 6.73i)T \)
good2 \( 1 + (0.410 - 0.711i)T + (-2 - 3.46i)T^{2} \)
3 \( 1 + (-0.507 + 0.292i)T + (4.5 - 7.79i)T^{2} \)
11 \( 1 + (6.95 + 12.0i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 18.3iT - 169T^{2} \)
17 \( 1 + (8.98 - 5.19i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-24.4 - 14.1i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (4.84 - 8.38i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 17.7T + 841T^{2} \)
31 \( 1 + (-25.1 + 14.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (17.2 - 29.8i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 13.3iT - 1.68e3T^{2} \)
43 \( 1 - 0.607T + 1.84e3T^{2} \)
47 \( 1 + (5.51 + 3.18i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-47.1 - 81.6i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (3.99 - 2.30i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (8.66 + 5.00i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-14.1 - 24.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 115.T + 5.04e3T^{2} \)
73 \( 1 + (54.9 - 31.7i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-2.17 + 3.75i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 126. iT - 6.88e3T^{2} \)
89 \( 1 + (68.6 + 39.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 15.7iT - 9.40e3T^{2} \)
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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

−16.58227487440501126955189393288, −15.49455301283075311500725112176, −13.91752253763868219470582884955, −13.13717290683519792987990987993, −11.38490880448667285674115849005, −10.34893759518199082803466638385, −8.286019871991628305193384179042, −7.55643881898413433369925640323, −5.72381244159281797793558866364, −3.12022686675335999537069427628, 2.28033796020376678385424390076, 5.19518137808725939525715956849, 6.72662201413702129019351906833, 8.997248868972017236149680486490, 9.708632512621673348720600626462, 11.39187835277940963432999716898, 12.24258743839506168685417354243, 14.03689407002175470606074074417, 15.07307844247600091917722930689, 15.94698236442094091583712017932

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