↑

Properties

Label 2-35-5.2-c4-0-6
Degree $2$
Conductor $35$
Sign $0.836 + 0.547i$
Analytic cond. $3.61794$
Root an. cond. $1.90209$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.48 + 1.48i)2-s + (2.63 − 2.63i)3-s − 11.5i·4-s + (8.51 − 23.5i)5-s + 7.83·6-s + (13.0 + 13.0i)7-s + (40.9 − 40.9i)8-s + 67.0i·9-s + (47.5 − 22.2i)10-s + 18.0·11-s + (−30.6 − 30.6i)12-s + (−1.78 + 1.78i)13-s + 38.8i·14-s + (−39.5 − 84.4i)15-s − 64.0·16-s + (8.06 + 8.06i)17-s + ⋯
L(s)  = 1  + (0.370 + 0.370i)2-s + (0.293 − 0.293i)3-s − 0.724i·4-s + (0.340 − 0.940i)5-s + 0.217·6-s + (0.267 + 0.267i)7-s + (0.639 − 0.639i)8-s + 0.828i·9-s + (0.475 − 0.222i)10-s + 0.149·11-s + (−0.212 − 0.212i)12-s + (−0.0105 + 0.0105i)13-s + 0.198i·14-s + (−0.175 − 0.375i)15-s − 0.250·16-s + (0.0278 + 0.0278i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.836 + 0.547i$
Analytic conductor: \(3.61794\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :2),\ 0.836 + 0.547i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.87566 - 0.558966i\)
\(L(\frac12)\) \(\approx\) \(1.87566 - 0.558966i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-8.51 + 23.5i)T \)
7 \( 1 + (-13.0 - 13.0i)T \)
good2 \( 1 + (-1.48 - 1.48i)T + 16iT^{2} \)
3 \( 1 + (-2.63 + 2.63i)T - 81iT^{2} \)
11 \( 1 - 18.0T + 1.46e4T^{2} \)
13 \( 1 + (1.78 - 1.78i)T - 2.85e4iT^{2} \)
17 \( 1 + (-8.06 - 8.06i)T + 8.35e4iT^{2} \)
19 \( 1 - 201. iT - 1.30e5T^{2} \)
23 \( 1 + (414. - 414. i)T - 2.79e5iT^{2} \)
29 \( 1 - 1.06e3iT - 7.07e5T^{2} \)
31 \( 1 - 1.36e3T + 9.23e5T^{2} \)
37 \( 1 + (848. + 848. i)T + 1.87e6iT^{2} \)
41 \( 1 - 2.80e3T + 2.82e6T^{2} \)
43 \( 1 + (-322. + 322. i)T - 3.41e6iT^{2} \)
47 \( 1 + (907. + 907. i)T + 4.87e6iT^{2} \)
53 \( 1 + (3.08e3 - 3.08e3i)T - 7.89e6iT^{2} \)
59 \( 1 - 1.71e3iT - 1.21e7T^{2} \)
61 \( 1 + 4.95e3T + 1.38e7T^{2} \)
67 \( 1 + (5.32e3 + 5.32e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 3.84e3T + 2.54e7T^{2} \)
73 \( 1 + (-3.29e3 + 3.29e3i)T - 2.83e7iT^{2} \)
79 \( 1 + 9.89e3iT - 3.89e7T^{2} \)
83 \( 1 + (-5.25e3 + 5.25e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 3.69e3iT - 6.27e7T^{2} \)
97 \( 1 + (9.93e3 + 9.93e3i)T + 8.85e7iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.67779741877390853072432179461, −14.26448862762372943758048076260, −13.56802070238084974522265297042, −12.32067841881945338187950040828, −10.59136706210824922463805456186, −9.214808739726931152649989935366, −7.77260471045293473070919763575, −5.92358256527821394067362025116, −4.73106711386405889364782755254, −1.62079248671558253507526855930, 2.74448704425941585754111077733, 4.18253553102180810689646077794, 6.54385927136482023194223201673, 8.085548855351263917663769719546, 9.719859548699209530982191820647, 11.08556721181966780751495034190, 12.18729557956347382187014002775, 13.62200068899704184564507742276, 14.49064363128001768924800494317, 15.71366667729087330520229362616

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