Properties

Label 2-35-7.3-c2-0-2
Degree $2$
Conductor $35$
Sign $0.754 + 0.655i$
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  + (−1.18 − 2.04i)2-s + (4.45 + 2.57i)3-s + (−0.796 + 1.37i)4-s + (−1.93 + 1.11i)5-s − 12.1i·6-s + (−1.42 − 6.85i)7-s − 5.69·8-s + (8.72 + 15.1i)9-s + (4.57 + 2.64i)10-s + (−6.86 + 11.8i)11-s + (−7.09 + 4.09i)12-s − 6.14i·13-s + (−12.3 + 11.0i)14-s − 11.5·15-s + (9.91 + 17.1i)16-s + (−1.74 − 1.00i)17-s + ⋯
L(s)  = 1  + (−0.591 − 1.02i)2-s + (1.48 + 0.857i)3-s + (−0.199 + 0.344i)4-s + (−0.387 + 0.223i)5-s − 2.02i·6-s + (−0.203 − 0.979i)7-s − 0.711·8-s + (0.969 + 1.67i)9-s + (0.457 + 0.264i)10-s + (−0.624 + 1.08i)11-s + (−0.591 + 0.341i)12-s − 0.472i·13-s + (−0.882 + 0.787i)14-s − 0.766·15-s + (0.619 + 1.07i)16-s + (−0.102 − 0.0591i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 + 0.655i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.754 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.754 + 0.655i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ 0.754 + 0.655i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.997874 - 0.372992i\)
\(L(\frac12)\) \(\approx\) \(0.997874 - 0.372992i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.93 - 1.11i)T \)
7 \( 1 + (1.42 + 6.85i)T \)
good2 \( 1 + (1.18 + 2.04i)T + (-2 + 3.46i)T^{2} \)
3 \( 1 + (-4.45 - 2.57i)T + (4.5 + 7.79i)T^{2} \)
11 \( 1 + (6.86 - 11.8i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 6.14iT - 169T^{2} \)
17 \( 1 + (1.74 + 1.00i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (2.08 - 1.20i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (8.02 + 13.9i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 15.9T + 841T^{2} \)
31 \( 1 + (-17.9 - 10.3i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (6.51 + 11.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 55.2iT - 1.68e3T^{2} \)
43 \( 1 - 54.7T + 1.84e3T^{2} \)
47 \( 1 + (-61.9 + 35.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (33.6 - 58.3i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-18.2 - 10.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (22.8 - 13.2i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (37.2 - 64.4i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 66.7T + 5.04e3T^{2} \)
73 \( 1 + (108. + 62.5i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-49.0 - 84.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 78.1iT - 6.88e3T^{2} \)
89 \( 1 + (-38.7 + 22.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 31.6iT - 9.40e3T^{2} \)
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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

−15.86832825571677894617864867120, −15.05586112155304423935224705272, −13.92018798203375517813423252904, −12.48394428322147583735450823148, −10.54922505814661445300794235199, −10.15882544616813418280657533652, −8.874027093163730894896327483706, −7.53902559562379236838421407041, −4.13109165277238082418957881702, −2.65955873603944789062487385177, 2.93283753868484626913008327884, 6.20692470184162015125425576547, 7.68782949341252012548212798650, 8.466003432342307160343984198624, 9.274940883569209300947278853422, 11.90577481553849794158658955645, 13.12450318649797162887209471385, 14.33646751744319511750288682048, 15.41634750643584501017140297340, 16.15115225547608644693268225864

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