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Properties

Label 2-35-7.2-c5-0-7
Degree $2$
Conductor $35$
Sign $0.999 + 0.00443i$
Analytic cond. $5.61343$
Root an. cond. $2.36926$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.36 + 7.55i)2-s + (−2.36 − 4.10i)3-s + (−22.0 − 38.2i)4-s + (12.5 − 21.6i)5-s + 41.3·6-s + (8.78 − 129. i)7-s + 105.·8-s + (110. − 190. i)9-s + (109. + 188. i)10-s + (339. + 588. i)11-s + (−104. + 181. i)12-s + 58.8·13-s + (939. + 630. i)14-s − 118.·15-s + (244. − 422. i)16-s + (−713. − 1.23e3i)17-s + ⋯
L(s)  = 1  + (−0.771 + 1.33i)2-s + (−0.152 − 0.263i)3-s + (−0.689 − 1.19i)4-s + (0.223 − 0.387i)5-s + 0.468·6-s + (0.0677 − 0.997i)7-s + 0.584·8-s + (0.453 − 0.785i)9-s + (0.344 + 0.597i)10-s + (0.846 + 1.46i)11-s + (−0.209 + 0.363i)12-s + 0.0966·13-s + (1.28 + 0.859i)14-s − 0.135·15-s + (0.238 − 0.413i)16-s + (−0.599 − 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00443i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.999 + 0.00443i$
Analytic conductor: \(5.61343\)
Root analytic conductor: \(2.36926\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :5/2),\ 0.999 + 0.00443i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.949921 - 0.00210661i\)
\(L(\frac12)\) \(\approx\) \(0.949921 - 0.00210661i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-12.5 + 21.6i)T \)
7 \( 1 + (-8.78 + 129. i)T \)
good2 \( 1 + (4.36 - 7.55i)T + (-16 - 27.7i)T^{2} \)
3 \( 1 + (2.36 + 4.10i)T + (-121.5 + 210. i)T^{2} \)
11 \( 1 + (-339. - 588. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 58.8T + 3.71e5T^{2} \)
17 \( 1 + (713. + 1.23e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-1.09e3 + 1.90e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-1.11e3 + 1.92e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 2.07e3T + 2.05e7T^{2} \)
31 \( 1 + (1.79e3 + 3.11e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (4.59e3 - 7.95e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 1.00e4T + 1.15e8T^{2} \)
43 \( 1 + 1.39e4T + 1.47e8T^{2} \)
47 \( 1 + (-8.13e3 + 1.40e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (633. + 1.09e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-6.73e3 - 1.16e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.75e4 - 3.03e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.84e4 - 3.19e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 4.11e3T + 1.80e9T^{2} \)
73 \( 1 + (-1.75e3 - 3.04e3i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-3.62e4 + 6.27e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 4.22e4T + 3.93e9T^{2} \)
89 \( 1 + (4.72e4 - 8.18e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 8.52e4T + 8.58e9T^{2} \)
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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

−15.68661988374316449325095111339, −14.70802331541504049941051487924, −13.38390197820975139267404140240, −11.90168182978869778474149355312, −9.885613542771344312798531053828, −9.030501016725955530384169763157, −7.23339983770794620082898291523, −6.73012800496113466410593076840, −4.67358535732919136176519087688, −0.808354729046706532640574935119, 1.69322935070872099858091743813, 3.44244152636642292982233710761, 5.89022428875269206436730112247, 8.311816167053148983934812019794, 9.400129211830444986381297547412, 10.68118807723599078921267876735, 11.43827192256103571550322371227, 12.66835487013206937260050758971, 14.13280170592537011425507670128, 15.70909499322149834104691916001

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