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Properties

Label 2-35-5.3-c4-0-9
Degree $2$
Conductor $35$
Sign $0.645 + 0.764i$
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  + (5.12 − 5.12i)2-s + (10.3 + 10.3i)3-s − 36.5i·4-s + (−22.2 + 11.3i)5-s + 106.·6-s + (13.0 − 13.0i)7-s + (−105. − 105. i)8-s + 132. i·9-s + (−56.3 + 172. i)10-s − 135.·11-s + (378. − 378. i)12-s + (−35.4 − 35.4i)13-s − 134. i·14-s + (−347. − 113. i)15-s − 497.·16-s + (146. − 146. i)17-s + ⋯
L(s)  = 1  + (1.28 − 1.28i)2-s + (1.14 + 1.14i)3-s − 2.28i·4-s + (−0.891 + 0.452i)5-s + 2.94·6-s + (0.267 − 0.267i)7-s + (−1.64 − 1.64i)8-s + 1.63i·9-s + (−0.563 + 1.72i)10-s − 1.12·11-s + (2.62 − 2.62i)12-s + (−0.209 − 0.209i)13-s − 0.685i·14-s + (−1.54 − 0.505i)15-s − 1.94·16-s + (0.506 − 0.506i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 + 0.764i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.645 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.645 + 0.764i$
Analytic conductor: \(3.61794\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :2),\ 0.645 + 0.764i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.76646 - 1.28512i\)
\(L(\frac12)\) \(\approx\) \(2.76646 - 1.28512i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (22.2 - 11.3i)T \)
7 \( 1 + (-13.0 + 13.0i)T \)
good2 \( 1 + (-5.12 + 5.12i)T - 16iT^{2} \)
3 \( 1 + (-10.3 - 10.3i)T + 81iT^{2} \)
11 \( 1 + 135.T + 1.46e4T^{2} \)
13 \( 1 + (35.4 + 35.4i)T + 2.85e4iT^{2} \)
17 \( 1 + (-146. + 146. i)T - 8.35e4iT^{2} \)
19 \( 1 - 408. iT - 1.30e5T^{2} \)
23 \( 1 + (-212. - 212. i)T + 2.79e5iT^{2} \)
29 \( 1 + 195. iT - 7.07e5T^{2} \)
31 \( 1 - 965.T + 9.23e5T^{2} \)
37 \( 1 + (-925. + 925. i)T - 1.87e6iT^{2} \)
41 \( 1 + 2.29e3T + 2.82e6T^{2} \)
43 \( 1 + (1.19e3 + 1.19e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (1.55e3 - 1.55e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (-645. - 645. i)T + 7.89e6iT^{2} \)
59 \( 1 + 2.29e3iT - 1.21e7T^{2} \)
61 \( 1 - 3.19e3T + 1.38e7T^{2} \)
67 \( 1 + (6.11e3 - 6.11e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 740.T + 2.54e7T^{2} \)
73 \( 1 + (-1.76e3 - 1.76e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 8.40e3iT - 3.89e7T^{2} \)
83 \( 1 + (-5.89e3 - 5.89e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 4.76e3iT - 6.27e7T^{2} \)
97 \( 1 + (-687. + 687. i)T - 8.85e7iT^{2} \)
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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

−15.08774897844104608655175847464, −14.41871974633860725295671511245, −13.37443529975830038891117395674, −11.90557643836239042134820637237, −10.65136761782133181312890402627, −9.934836689424456716971172669107, −7.995651048223811574529354971223, −5.01517858982937958699127294868, −3.78852929502642859517961430017, −2.78360456896506542987146549569, 3.03799191838971872083692528809, 4.87919997230110402814278682051, 6.81142141861829439517942163454, 7.87964296141236243881487076467, 8.510560966426606922087736402470, 11.91547538747760195265017504395, 12.91975921853151144167890542894, 13.54138233588021185887294117493, 14.84059829525197620377059918099, 15.39059065056413138043054693781

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