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Properties

Label 2-35-7.6-c4-0-6
Degree $2$
Conductor $35$
Sign $0.913 - 0.407i$
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  + 6.50·2-s + 5.52i·3-s + 26.2·4-s + 11.1i·5-s + 35.9i·6-s + (−19.9 − 44.7i)7-s + 66.9·8-s + 50.4·9-s + 72.7i·10-s − 105.·11-s + 145. i·12-s − 234. i·13-s + (−129. − 290. i)14-s − 61.8·15-s + 14.4·16-s + 375. i·17-s + ⋯
L(s)  = 1  + 1.62·2-s + 0.614i·3-s + 1.64·4-s + 0.447i·5-s + 0.998i·6-s + (−0.407 − 0.913i)7-s + 1.04·8-s + 0.622·9-s + 0.727i·10-s − 0.871·11-s + 1.00i·12-s − 1.38i·13-s + (−0.663 − 1.48i)14-s − 0.274·15-s + 0.0565·16-s + 1.29i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.913 - 0.407i$
Analytic conductor: \(3.61794\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :2),\ 0.913 - 0.407i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.97529 + 0.634480i\)
\(L(\frac12)\) \(\approx\) \(2.97529 + 0.634480i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 11.1iT \)
7 \( 1 + (19.9 + 44.7i)T \)
good2 \( 1 - 6.50T + 16T^{2} \)
3 \( 1 - 5.52iT - 81T^{2} \)
11 \( 1 + 105.T + 1.46e4T^{2} \)
13 \( 1 + 234. iT - 2.85e4T^{2} \)
17 \( 1 - 375. iT - 8.35e4T^{2} \)
19 \( 1 + 45.1iT - 1.30e5T^{2} \)
23 \( 1 - 46.3T + 2.79e5T^{2} \)
29 \( 1 + 257.T + 7.07e5T^{2} \)
31 \( 1 - 194. iT - 9.23e5T^{2} \)
37 \( 1 - 2.52e3T + 1.87e6T^{2} \)
41 \( 1 - 2.50e3iT - 2.82e6T^{2} \)
43 \( 1 - 2.28e3T + 3.41e6T^{2} \)
47 \( 1 + 2.62e3iT - 4.87e6T^{2} \)
53 \( 1 + 358.T + 7.89e6T^{2} \)
59 \( 1 + 854. iT - 1.21e7T^{2} \)
61 \( 1 - 2.08e3iT - 1.38e7T^{2} \)
67 \( 1 + 7.55e3T + 2.01e7T^{2} \)
71 \( 1 + 282.T + 2.54e7T^{2} \)
73 \( 1 + 9.19e3iT - 2.83e7T^{2} \)
79 \( 1 + 7.18e3T + 3.89e7T^{2} \)
83 \( 1 - 5.06e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.02e4iT - 6.27e7T^{2} \)
97 \( 1 - 608. iT - 8.85e7T^{2} \)
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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

−15.41409435094372122290288648881, −14.77861648086314111032840575300, −13.27016469961574792042427397147, −12.80408653072300554250382924207, −10.92620808530375258836954659076, −10.14611077776341814374060476140, −7.53804239992908880992602428465, −5.97916444401860471930866079693, −4.44024724820835891657827730119, −3.19660428043204240013767427160, 2.42810823308654331708997695685, 4.50940102440545341903793585334, 5.90816234543533870065787926681, 7.26557453944566111329480644294, 9.326222293082703930694301601744, 11.50402598433125393109930671371, 12.44032853948605449536536318738, 13.18998660299844710793783304690, 14.19127122431335704297034940427, 15.60082015011939231478816911148

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