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Properties

Label 2-35-35.4-c3-0-4
Degree $2$
Conductor $35$
Sign $0.907 - 0.420i$
Analytic cond. $2.06506$
Root an. cond. $1.43703$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.736 − 0.425i)2-s + (3.23 + 1.86i)3-s + (−3.63 + 6.30i)4-s + (10.5 + 3.83i)5-s + 3.17·6-s + (3.68 − 18.1i)7-s + 12.9i·8-s + (−6.52 − 11.2i)9-s + (9.36 − 1.64i)10-s + (−4.34 + 7.52i)11-s + (−23.5 + 13.5i)12-s + 0.148i·13-s + (−5.00 − 14.9i)14-s + (26.8 + 32.0i)15-s + (−23.5 − 40.8i)16-s + (−102. − 58.9i)17-s + ⋯
L(s)  = 1  + (0.260 − 0.150i)2-s + (0.622 + 0.359i)3-s + (−0.454 + 0.787i)4-s + (0.939 + 0.342i)5-s + 0.216·6-s + (0.198 − 0.980i)7-s + 0.574i·8-s + (−0.241 − 0.418i)9-s + (0.296 − 0.0519i)10-s + (−0.119 + 0.206i)11-s + (−0.566 + 0.326i)12-s + 0.00317i·13-s + (−0.0955 − 0.285i)14-s + (0.461 + 0.551i)15-s + (−0.368 − 0.638i)16-s + (−1.45 − 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.907 - 0.420i$
Analytic conductor: \(2.06506\)
Root analytic conductor: \(1.43703\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :3/2),\ 0.907 - 0.420i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.61705 + 0.356176i\)
\(L(\frac12)\) \(\approx\) \(1.61705 + 0.356176i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-10.5 - 3.83i)T \)
7 \( 1 + (-3.68 + 18.1i)T \)
good2 \( 1 + (-0.736 + 0.425i)T + (4 - 6.92i)T^{2} \)
3 \( 1 + (-3.23 - 1.86i)T + (13.5 + 23.3i)T^{2} \)
11 \( 1 + (4.34 - 7.52i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 0.148iT - 2.19e3T^{2} \)
17 \( 1 + (102. + 58.9i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-10.2 - 17.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-56.3 + 32.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 211.T + 2.43e4T^{2} \)
31 \( 1 + (139. - 241. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-53.8 + 31.0i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 93.6T + 6.89e4T^{2} \)
43 \( 1 - 346. iT - 7.95e4T^{2} \)
47 \( 1 + (193. - 111. i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (418. + 241. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (91.8 - 159. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-185. - 320. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-903. - 521. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 478.T + 3.57e5T^{2} \)
73 \( 1 + (261. + 151. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-387. - 671. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 83.8iT - 5.71e5T^{2} \)
89 \( 1 + (524. + 908. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 164. iT - 9.12e5T^{2} \)
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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

−16.18408237815293300741628879005, −14.50175152433351752967812740618, −13.86282738299652076056958619691, −12.84343240237793136836557296203, −11.15575739456891595881162930911, −9.704019467340451987324740336074, −8.588497787348023316091997971451, −6.90399547250982426515666128180, −4.60680088653706771368773099549, −2.95833639037515688542326756396, 2.10190454842051387923174120043, 5.03280875284189849234316493241, 6.24917724158989762218158407844, 8.502084288631232184621757061033, 9.360364163521278919639493107469, 10.90278774883847926259222263254, 12.87775392497721223540060379608, 13.61040989886483891235133811189, 14.60834375805731829312816343785, 15.64029279909196416849070442524

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