Properties

Label 2-35-35.19-c4-0-6
Degree $2$
Conductor $35$
Sign $0.592 - 0.805i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.18 + 2.99i)2-s + (−1.73 − 2.99i)3-s + (9.90 + 17.1i)4-s + (15.6 + 19.5i)5-s − 20.7i·6-s + (18.5 + 45.3i)7-s + 22.8i·8-s + (34.5 − 59.7i)9-s + (22.6 + 147. i)10-s + (−90.7 − 157. i)11-s + (34.2 − 59.3i)12-s − 258.·13-s + (−39.6 + 290. i)14-s + (31.4 − 80.6i)15-s + (90.1 − 156. i)16-s + (−21.6 − 37.4i)17-s + ⋯
L(s)  = 1  + (1.29 + 0.748i)2-s + (−0.192 − 0.333i)3-s + (0.619 + 1.07i)4-s + (0.625 + 0.780i)5-s − 0.575i·6-s + (0.378 + 0.925i)7-s + 0.356i·8-s + (0.426 − 0.737i)9-s + (0.226 + 1.47i)10-s + (−0.749 − 1.29i)11-s + (0.238 − 0.412i)12-s − 1.52·13-s + (−0.202 + 1.48i)14-s + (0.139 − 0.358i)15-s + (0.352 − 0.610i)16-s + (−0.0749 − 0.129i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.592 - 0.805i$
Analytic conductor: \(3.61794\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :2),\ 0.592 - 0.805i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.36939 + 1.19875i\)
\(L(\frac12)\) \(\approx\) \(2.36939 + 1.19875i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-15.6 - 19.5i)T \)
7 \( 1 + (-18.5 - 45.3i)T \)
good2 \( 1 + (-5.18 - 2.99i)T + (8 + 13.8i)T^{2} \)
3 \( 1 + (1.73 + 2.99i)T + (-40.5 + 70.1i)T^{2} \)
11 \( 1 + (90.7 + 157. i)T + (-7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + 258.T + 2.85e4T^{2} \)
17 \( 1 + (21.6 + 37.4i)T + (-4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-161. - 93.5i)T + (6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (-374. - 216. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + 647.T + 7.07e5T^{2} \)
31 \( 1 + (-540. + 312. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (-282. - 162. i)T + (9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 - 2.88e3iT - 2.82e6T^{2} \)
43 \( 1 + 1.29e3iT - 3.41e6T^{2} \)
47 \( 1 + (-177. + 307. i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (796. - 459. i)T + (3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (1.85e3 - 1.07e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (2.37e3 + 1.37e3i)T + (6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (3.44e3 - 1.98e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 7.72e3T + 2.54e7T^{2} \)
73 \( 1 + (-931. - 1.61e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (1.99e3 - 3.45e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 6.81e3T + 4.74e7T^{2} \)
89 \( 1 + (1.33e3 + 769. i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + 1.29e4T + 8.85e7T^{2} \)
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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

−15.44081494315685262550336799788, −14.79173749382159256506276282039, −13.71773387798631573231863549259, −12.66419425341830412272921757872, −11.49568155593995451736777698524, −9.646261299389547133975026420392, −7.52156744863728778438588471151, −6.22067610043553655176518966729, −5.24538747299408177026323592946, −2.96121635211646215674310572015, 2.09614284956682936489680121481, 4.59439970290687948182998765093, 5.06321930741411014519847624088, 7.50098390283616329168875149339, 9.837480000545880931864807405974, 10.73962630094029329724393488989, 12.31699525348064335149736962166, 13.08448300655746537008794928948, 14.07134702852547499674528733038, 15.26069269271568498151578586011

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