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Properties

Label 2-35-7.2-c5-0-5
Degree $2$
Conductor $35$
Sign $-0.153 - 0.988i$
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  + (−0.175 + 0.303i)2-s + (10.7 + 18.6i)3-s + (15.9 + 27.6i)4-s + (12.5 − 21.6i)5-s − 7.56·6-s + (−129. + 11.8i)7-s − 22.3·8-s + (−111. + 193. i)9-s + (4.38 + 7.58i)10-s + (96.5 + 167. i)11-s + (−344. + 596. i)12-s + 691.·13-s + (19.0 − 41.2i)14-s + 539.·15-s + (−506. + 876. i)16-s + (−324. − 562. i)17-s + ⋯
L(s)  = 1  + (−0.0309 + 0.0536i)2-s + (0.692 + 1.19i)3-s + (0.498 + 0.862i)4-s + (0.223 − 0.387i)5-s − 0.0858·6-s + (−0.995 + 0.0910i)7-s − 0.123·8-s + (−0.459 + 0.795i)9-s + (0.0138 + 0.0239i)10-s + (0.240 + 0.416i)11-s + (−0.689 + 1.19i)12-s + 1.13·13-s + (0.0259 − 0.0562i)14-s + 0.619·15-s + (−0.494 + 0.856i)16-s + (−0.272 − 0.472i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.153 - 0.988i$
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: \((2,\ 35,\ (\ :5/2),\ -0.153 - 0.988i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.28592 + 1.50176i\)
\(L(\frac12)\) \(\approx\) \(1.28592 + 1.50176i\)
\(L(\frac{7}{2})\) 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 \( 1 + (129. - 11.8i)T \)
good2 \( 1 + (0.175 - 0.303i)T + (-16 - 27.7i)T^{2} \)
3 \( 1 + (-10.7 - 18.6i)T + (-121.5 + 210. i)T^{2} \)
11 \( 1 + (-96.5 - 167. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 691.T + 3.71e5T^{2} \)
17 \( 1 + (324. + 562. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (348. - 603. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-1.48e3 + 2.58e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 6.77e3T + 2.05e7T^{2} \)
31 \( 1 + (75.9 + 131. i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-5.50e3 + 9.53e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 7.36e3T + 1.15e8T^{2} \)
43 \( 1 + 1.92e4T + 1.47e8T^{2} \)
47 \( 1 + (6.39e3 - 1.10e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (4.53e3 + 7.85e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.84e4 - 3.19e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-1.17e4 + 2.02e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (3.22e4 + 5.57e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 8.31e4T + 1.80e9T^{2} \)
73 \( 1 + (1.44e4 + 2.50e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (845. - 1.46e3i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 8.03e4T + 3.93e9T^{2} \)
89 \( 1 + (5.94e4 - 1.02e5i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 1.47e5T + 8.58e9T^{2} \)
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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

−16.12307783504506105108109269831, −14.95006766513991174800949537836, −13.43200917240777874503902173386, −12.29849150959172942106889752532, −10.66383163995823427234488325972, −9.365089965392690200921629256438, −8.443973113930171145869526708323, −6.52601645209655214978001309670, −4.22979508956640852539517525416, −2.94341102592246199675592476159, 1.26112916341533221831136156061, 2.94445297331607660502917381050, 6.16407338624007363274852011876, 6.92357802385578862117386088854, 8.660523576011378114062741249508, 10.14573277332573845204424782307, 11.50779898115351775539346417428, 13.14467723773651836029099103428, 13.76930708407092554198437646194, 15.03844297294111072169113485684

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