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Properties

Label 2-35-35.24-c4-0-6
Degree $2$
Conductor $35$
Sign $0.387 - 0.921i$
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  + (2.10 − 1.21i)2-s + (−3.73 + 6.46i)3-s + (−5.05 + 8.75i)4-s + (24.9 + 0.969i)5-s + 18.1i·6-s + (−22.9 + 43.3i)7-s + 63.3i·8-s + (12.6 + 21.8i)9-s + (53.6 − 28.2i)10-s + (95.0 − 164. i)11-s + (−37.7 − 65.3i)12-s − 63.2·13-s + (4.42 + 118. i)14-s + (−99.5 + 157. i)15-s + (−3.93 − 6.81i)16-s + (166. − 288. i)17-s + ⋯
L(s)  = 1  + (0.525 − 0.303i)2-s + (−0.414 + 0.718i)3-s + (−0.315 + 0.547i)4-s + (0.999 + 0.0387i)5-s + 0.503i·6-s + (−0.467 + 0.884i)7-s + 0.990i·8-s + (0.155 + 0.269i)9-s + (0.536 − 0.282i)10-s + (0.785 − 1.36i)11-s + (−0.262 − 0.453i)12-s − 0.374·13-s + (0.0225 + 0.606i)14-s + (−0.442 + 0.701i)15-s + (−0.0153 − 0.0266i)16-s + (0.575 − 0.996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.387 - 0.921i$
Analytic conductor: \(3.61794\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :2),\ 0.387 - 0.921i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.38106 + 0.917081i\)
\(L(\frac12)\) \(\approx\) \(1.38106 + 0.917081i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-24.9 - 0.969i)T \)
7 \( 1 + (22.9 - 43.3i)T \)
good2 \( 1 + (-2.10 + 1.21i)T + (8 - 13.8i)T^{2} \)
3 \( 1 + (3.73 - 6.46i)T + (-40.5 - 70.1i)T^{2} \)
11 \( 1 + (-95.0 + 164. i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + 63.2T + 2.85e4T^{2} \)
17 \( 1 + (-166. + 288. i)T + (-4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (343. - 198. i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (-361. + 208. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 - 1.38e3T + 7.07e5T^{2} \)
31 \( 1 + (-416. - 240. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (-1.36e3 + 788. i)T + (9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 - 959. iT - 2.82e6T^{2} \)
43 \( 1 - 1.68e3iT - 3.41e6T^{2} \)
47 \( 1 + (868. + 1.50e3i)T + (-2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (2.06e3 + 1.19e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-131. - 76.0i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (2.35e3 - 1.35e3i)T + (6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (2.68e3 + 1.55e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 739.T + 2.54e7T^{2} \)
73 \( 1 + (-2.97e3 + 5.14e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (-1.71e3 - 2.97e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + 1.09e4T + 4.74e7T^{2} \)
89 \( 1 + (2.30e3 - 1.32e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 - 5.79e3T + 8.85e7T^{2} \)
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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

−16.31535057801916794923150535398, −14.56815980879369491058878665088, −13.56967334490943708880807889087, −12.41028411518362126154898509546, −11.19793026217906361828503814059, −9.755139342935797916854682715984, −8.591145767454780082730835820035, −6.09928904405309615109305698626, −4.83254019556949706795551885020, −2.91371132365876966185024691474, 1.24977980630715878777546876687, 4.46559872925799341752749758531, 6.24177634367518227046259212571, 6.93989829092536929402857318920, 9.486603127066566250854368793315, 10.32995797954623831131681505419, 12.45757524276265878438991888722, 13.13128372727279307389923576969, 14.26104019518873256644486440368, 15.24030646332719122527305118456

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