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Properties

Label 2-35-7.4-c5-0-3
Degree $2$
Conductor $35$
Sign $-0.999 - 0.0172i$
Analytic cond. $5.61343$
Root an. cond. $2.36926$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.58 + 7.94i)2-s + (−7.16 + 12.4i)3-s + (−26.0 + 45.1i)4-s + (12.5 + 21.6i)5-s − 131.·6-s + (−5.98 − 129. i)7-s − 184.·8-s + (18.8 + 32.7i)9-s + (−114. + 198. i)10-s + (−107. + 186. i)11-s + (−373. − 646. i)12-s + 132.·13-s + (1.00e3 − 641. i)14-s − 358.·15-s + (−12.8 − 22.3i)16-s + (−220. + 382. i)17-s + ⋯
L(s)  = 1  + (0.810 + 1.40i)2-s + (−0.459 + 0.795i)3-s + (−0.814 + 1.41i)4-s + (0.223 + 0.387i)5-s − 1.49·6-s + (−0.0461 − 0.998i)7-s − 1.02·8-s + (0.0777 + 0.134i)9-s + (−0.362 + 0.628i)10-s + (−0.268 + 0.464i)11-s + (−0.748 − 1.29i)12-s + 0.217·13-s + (1.36 − 0.874i)14-s − 0.410·15-s + (−0.0125 − 0.0217i)16-s + (−0.185 + 0.320i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0172i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.999 - 0.0172i$
Analytic conductor: \(5.61343\)
Root analytic conductor: \(2.36926\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :5/2),\ -0.999 - 0.0172i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0168246 + 1.95482i\)
\(L(\frac12)\) \(\approx\) \(0.0168246 + 1.95482i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-12.5 - 21.6i)T \)
7 \( 1 + (5.98 + 129. i)T \)
good2 \( 1 + (-4.58 - 7.94i)T + (-16 + 27.7i)T^{2} \)
3 \( 1 + (7.16 - 12.4i)T + (-121.5 - 210. i)T^{2} \)
11 \( 1 + (107. - 186. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 132.T + 3.71e5T^{2} \)
17 \( 1 + (220. - 382. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (259. + 449. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-2.19e3 - 3.80e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 7.11e3T + 2.05e7T^{2} \)
31 \( 1 + (-5.15e3 + 8.92e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (2.78e3 + 4.82e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 7.24e3T + 1.15e8T^{2} \)
43 \( 1 + 184.T + 1.47e8T^{2} \)
47 \( 1 + (8.08e3 + 1.40e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (3.63e3 - 6.30e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-2.26e4 + 3.92e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-3.68e3 - 6.38e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-3.22e4 + 5.58e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 343.T + 1.80e9T^{2} \)
73 \( 1 + (2.26e4 - 3.92e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-3.72e4 - 6.45e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 5.47e4T + 3.93e9T^{2} \)
89 \( 1 + (-5.34e4 - 9.25e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.93e4T + 8.58e9T^{2} \)
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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

−15.86650266631252038438030205936, −15.19575659133049965218176583913, −13.90823410676383841650061231477, −13.09865233522738326217191797168, −11.07030503651489740694997670741, −9.895769007587317954557927547923, −7.80673963491774405828663901688, −6.63261566218854090742574195925, −5.16261644802413024715520346673, −4.01414038353882174150083577806, 1.06400642660563678510232127528, 2.75065863887597125188338686509, 4.89142556381375856183610890628, 6.33530653145746112421065810171, 8.713728425065174610502886571502, 10.32899833464399954587385269216, 11.69048812977634622190135853457, 12.41587757055696387509857721244, 13.16341294455732577546414560441, 14.40730021464397015180077803160

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