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Properties

Label 2-35-35.34-c2-0-0
Degree $2$
Conductor $35$
Sign $-0.692 - 0.721i$
Analytic cond. $0.953680$
Root an. cond. $0.976565$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·2-s − 3.16·3-s − 5·4-s + (1.58 + 4.74i)5-s − 9.48i·6-s + (6.32 − 3i)7-s − 3i·8-s + 1.00·9-s + (−14.2 + 4.74i)10-s + 14·11-s + 15.8·12-s + 3.16·13-s + (9 + 18.9i)14-s + (−5.00 − 15.0i)15-s − 11·16-s − 6.32·17-s + ⋯
L(s)  = 1  + 1.5i·2-s − 1.05·3-s − 1.25·4-s + (0.316 + 0.948i)5-s − 1.58i·6-s + (0.903 − 0.428i)7-s − 0.375i·8-s + 0.111·9-s + (−1.42 + 0.474i)10-s + 1.27·11-s + 1.31·12-s + 0.243·13-s + (0.642 + 1.35i)14-s + (−0.333 − i)15-s − 0.687·16-s − 0.372·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.692 - 0.721i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ -0.692 - 0.721i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.326401 + 0.765458i\)
\(L(\frac12)\) \(\approx\) \(0.326401 + 0.765458i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.58 - 4.74i)T \)
7 \( 1 + (-6.32 + 3i)T \)
good2 \( 1 - 3iT - 4T^{2} \)
3 \( 1 + 3.16T + 9T^{2} \)
11 \( 1 - 14T + 121T^{2} \)
13 \( 1 - 3.16T + 169T^{2} \)
17 \( 1 + 6.32T + 289T^{2} \)
19 \( 1 + 28.4iT - 361T^{2} \)
23 \( 1 + 12iT - 529T^{2} \)
29 \( 1 - 14T + 841T^{2} \)
31 \( 1 - 37.9iT - 961T^{2} \)
37 \( 1 - 18iT - 1.36e3T^{2} \)
41 \( 1 + 18.9iT - 1.68e3T^{2} \)
43 \( 1 + 42iT - 1.84e3T^{2} \)
47 \( 1 + 44.2T + 2.20e3T^{2} \)
53 \( 1 - 54iT - 2.80e3T^{2} \)
59 \( 1 - 9.48iT - 3.48e3T^{2} \)
61 \( 1 + 66.4iT - 3.72e3T^{2} \)
67 \( 1 + 102iT - 4.48e3T^{2} \)
71 \( 1 + 16T + 5.04e3T^{2} \)
73 \( 1 + 63.2T + 5.32e3T^{2} \)
79 \( 1 + 76T + 6.24e3T^{2} \)
83 \( 1 + 72.7T + 6.88e3T^{2} \)
89 \( 1 - 56.9iT - 7.92e3T^{2} \)
97 \( 1 - 69.5T + 9.40e3T^{2} \)
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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

−17.02079341486999203038148665300, −15.65631854950893192362428752794, −14.53917766918590550321504440888, −13.82682411173122446961331618186, −11.64517565664980042195073372917, −10.77547695345533703248418065152, −8.777850099212590674768029056044, −7.04964578408385943295610019383, −6.31938465392784357736936317211, −4.84772126649158131360788524174, 1.44973716394288436679186751702, 4.36093397936944276801177774641, 5.89416070913243298913157686627, 8.631793773264566698803235714984, 9.915596054724230591623728215155, 11.43611550519918476870411483327, 11.80332801492276377411359897480, 12.89255295045205744625222465377, 14.35871251949844546698359710515, 16.30393260043941570316373843547

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