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Properties

Label 2-35-7.3-c2-0-0
Degree $2$
Conductor $35$
Sign $0.128 - 0.991i$
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  + (1.68 + 2.92i)2-s + (−1.83 − 1.06i)3-s + (−3.70 + 6.42i)4-s + (1.93 − 1.11i)5-s − 7.16i·6-s + (4.91 − 4.98i)7-s − 11.5·8-s + (−2.25 − 3.89i)9-s + (6.54 + 3.77i)10-s + (−7.37 + 12.7i)11-s + (13.6 − 7.86i)12-s − 12.3i·13-s + (22.8 + 5.97i)14-s − 4.74·15-s + (−4.66 − 8.08i)16-s + (−2.62 − 1.51i)17-s + ⋯
L(s)  = 1  + (0.844 + 1.46i)2-s + (−0.612 − 0.353i)3-s + (−0.927 + 1.60i)4-s + (0.387 − 0.223i)5-s − 1.19i·6-s + (0.702 − 0.711i)7-s − 1.44·8-s + (−0.250 − 0.433i)9-s + (0.654 + 0.377i)10-s + (−0.670 + 1.16i)11-s + (1.13 − 0.655i)12-s − 0.949i·13-s + (1.63 + 0.427i)14-s − 0.316·15-s + (−0.291 − 0.505i)16-s + (−0.154 − 0.0890i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.128 - 0.991i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ 0.128 - 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.969792 + 0.852276i\)
\(L(\frac12)\) \(\approx\) \(0.969792 + 0.852276i\)
\(L(2)\) 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 \( 1 + (-4.91 + 4.98i)T \)
good2 \( 1 + (-1.68 - 2.92i)T + (-2 + 3.46i)T^{2} \)
3 \( 1 + (1.83 + 1.06i)T + (4.5 + 7.79i)T^{2} \)
11 \( 1 + (7.37 - 12.7i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 12.3iT - 169T^{2} \)
17 \( 1 + (2.62 + 1.51i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (22.5 - 13.0i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (1.54 + 2.67i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 53.5T + 841T^{2} \)
31 \( 1 + (24.5 + 14.1i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (1.08 + 1.88i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 42.3iT - 1.68e3T^{2} \)
43 \( 1 - 27.3T + 1.84e3T^{2} \)
47 \( 1 + (-44.6 + 25.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-8.99 + 15.5i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-19.1 - 11.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (12.6 - 7.32i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (7.55 - 13.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 58.9T + 5.04e3T^{2} \)
73 \( 1 + (38.6 + 22.3i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (9.00 + 15.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 72.1iT - 6.88e3T^{2} \)
89 \( 1 + (8.38 - 4.84i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 145. iT - 9.40e3T^{2} \)
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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

−16.62576292460471245416587873018, −15.25852275145785322612179429130, −14.45619060889446501071574097265, −13.16352041196357942547191791167, −12.33832444812898439929675000257, −10.44578568088254347992778978484, −8.261502577557115807430564913991, −7.07219005015304555021906895426, −5.79532006443254472280295179654, −4.54741080603320411772198223109, 2.45300743904323791433935688585, 4.64762650962743563804669559187, 5.81648850069236098402121624755, 8.773343212238341529444453665178, 10.57398492950362349221072105395, 11.10686642819388942971755200491, 12.15585774128752584423720538795, 13.54894406698359713874355407841, 14.36757920160920976771165436994, 15.93901882518870489469834349027

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