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Properties

Label 2-35-35.24-c2-0-4
Degree $2$
Conductor $35$
Sign $0.926 + 0.375i$
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  + (2.85 − 1.65i)2-s + (−2.20 + 3.81i)3-s + (3.44 − 5.97i)4-s + (−1.72 − 4.69i)5-s + 14.5i·6-s + (−5.32 + 4.54i)7-s − 9.55i·8-s + (−5.18 − 8.98i)9-s + (−12.6 − 10.5i)10-s + (−0.240 + 0.415i)11-s + (15.1 + 26.2i)12-s + 11.0·13-s + (−7.70 + 21.7i)14-s + (21.6 + 3.76i)15-s + (−1.98 − 3.42i)16-s + (9.10 − 15.7i)17-s + ⋯
L(s)  = 1  + (1.42 − 0.825i)2-s + (−0.733 + 1.27i)3-s + (0.861 − 1.49i)4-s + (−0.344 − 0.938i)5-s + 2.42i·6-s + (−0.760 + 0.649i)7-s − 1.19i·8-s + (−0.576 − 0.998i)9-s + (−1.26 − 1.05i)10-s + (−0.0218 + 0.0377i)11-s + (1.26 + 2.19i)12-s + 0.853·13-s + (−0.550 + 1.55i)14-s + (1.44 + 0.250i)15-s + (−0.123 − 0.214i)16-s + (0.535 − 0.927i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.926 + 0.375i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ 0.926 + 0.375i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.49725 - 0.291906i\)
\(L(\frac12)\) \(\approx\) \(1.49725 - 0.291906i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.72 + 4.69i)T \)
7 \( 1 + (5.32 - 4.54i)T \)
good2 \( 1 + (-2.85 + 1.65i)T + (2 - 3.46i)T^{2} \)
3 \( 1 + (2.20 - 3.81i)T + (-4.5 - 7.79i)T^{2} \)
11 \( 1 + (0.240 - 0.415i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 11.0T + 169T^{2} \)
17 \( 1 + (-9.10 + 15.7i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (5.12 - 2.95i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-8.07 + 4.66i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 10.3T + 841T^{2} \)
31 \( 1 + (1.63 + 0.945i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (42.5 - 24.5i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 11.9iT - 1.68e3T^{2} \)
43 \( 1 + 62.7iT - 1.84e3T^{2} \)
47 \( 1 + (-23.8 - 41.2i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-43.0 - 24.8i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (48.0 + 27.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (52.5 - 30.3i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (71.3 + 41.1i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 92.6T + 5.04e3T^{2} \)
73 \( 1 + (24.5 - 42.6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-1.72 - 2.98i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 30.5T + 6.88e3T^{2} \)
89 \( 1 + (-121. + 69.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 130.T + 9.40e3T^{2} \)
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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

−15.89557394736863704821939515314, −15.25527049668460409886801035960, −13.61361520783949889812225540421, −12.41931693281124383304787649431, −11.63605032323250610809841686060, −10.44624592561915534733061866689, −9.099359991437709829252933199551, −5.84280721159719062984368707852, −4.87164530388657982426852148911, −3.57223046117935976757880198671, 3.62588053964928696521340829576, 5.94181989789996030151930342281, 6.72037220101523387518030684258, 7.63848345525403943904857158726, 10.75126323600439902601537003533, 12.06687181364987015278771132663, 13.04252548777498096410234883437, 13.80623031074365241957122252157, 15.04092034269892028064490622424, 16.23501619543500510226859901932

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