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Properties

Label 2-35-35.19-c2-0-1
Degree $2$
Conductor $35$
Sign $0.853 - 0.521i$
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  + (0.226 + 0.130i)2-s + (1.88 + 3.25i)3-s + (−1.96 − 3.40i)4-s + (2.11 + 4.53i)5-s + 0.982i·6-s + (−1.66 − 6.80i)7-s − 2.07i·8-s + (−2.58 + 4.46i)9-s + (−0.114 + 1.30i)10-s + (−3.04 − 5.27i)11-s + (7.39 − 12.8i)12-s − 13.0·13-s + (0.512 − 1.75i)14-s + (−10.7 + 15.4i)15-s + (−7.59 + 13.1i)16-s + (−5.05 − 8.75i)17-s + ⋯
L(s)  = 1  + (0.113 + 0.0652i)2-s + (0.627 + 1.08i)3-s + (−0.491 − 0.851i)4-s + (0.422 + 0.906i)5-s + 0.163i·6-s + (−0.237 − 0.971i)7-s − 0.258i·8-s + (−0.286 + 0.496i)9-s + (−0.0114 + 0.130i)10-s + (−0.276 − 0.479i)11-s + (0.616 − 1.06i)12-s − 1.00·13-s + (0.0366 − 0.125i)14-s + (−0.719 + 1.02i)15-s + (−0.474 + 0.821i)16-s + (−0.297 − 0.514i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.853 - 0.521i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ 0.853 - 0.521i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.13139 + 0.318540i\)
\(L(\frac12)\) \(\approx\) \(1.13139 + 0.318540i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.11 - 4.53i)T \)
7 \( 1 + (1.66 + 6.80i)T \)
good2 \( 1 + (-0.226 - 0.130i)T + (2 + 3.46i)T^{2} \)
3 \( 1 + (-1.88 - 3.25i)T + (-4.5 + 7.79i)T^{2} \)
11 \( 1 + (3.04 + 5.27i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 13.0T + 169T^{2} \)
17 \( 1 + (5.05 + 8.75i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-19.5 - 11.2i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-30.2 - 17.4i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 16.3T + 841T^{2} \)
31 \( 1 + (23.4 - 13.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (14.4 + 8.33i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 19.7iT - 1.68e3T^{2} \)
43 \( 1 + 53.4iT - 1.84e3T^{2} \)
47 \( 1 + (18.3 - 31.7i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-31.9 + 18.4i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-92.4 + 53.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (18.4 + 10.6i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-5.33 + 3.07i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 48.1T + 5.04e3T^{2} \)
73 \( 1 + (2.82 + 4.88i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-19.1 + 33.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 102.T + 6.88e3T^{2} \)
89 \( 1 + (34.1 + 19.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 33.5T + 9.40e3T^{2} \)
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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

−16.14652426292994470133799031104, −14.98570228651617022627232430886, −14.28429189036380657381491166567, −13.42022115502608586585012649817, −10.95630517019648757234887828242, −10.02481217761782385766004558535, −9.328056506504269843707096545214, −7.12429344595222233553071624397, −5.20007916736036251412486373599, −3.47386711819906283513969281871, 2.49294430757270437422728954374, 5.05644031109521036720292130291, 7.24585474066281280603198813425, 8.492599489918824468645133241604, 9.390979953778334627997395803488, 12.01196235785321603216038031541, 12.82043328283220535358432099517, 13.35257951014206308480211081884, 14.80921405492287666469639871722, 16.43057609789267031420536433215

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