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Properties

Label 2-35-35.19-c4-0-5
Degree $2$
Conductor $35$
Sign $0.343 - 0.939i$
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  + (3.03 + 1.75i)2-s + (7.77 + 13.4i)3-s + (−1.87 − 3.24i)4-s + (20.1 − 14.8i)5-s + 54.4i·6-s + (−37.0 + 32.0i)7-s − 69.1i·8-s + (−80.3 + 139. i)9-s + (86.9 − 9.61i)10-s + (−47.6 − 82.4i)11-s + (29.1 − 50.4i)12-s + 194.·13-s + (−168. + 32.3i)14-s + (355. + 156. i)15-s + (91.0 − 157. i)16-s + (−117. − 203. i)17-s + ⋯
L(s)  = 1  + (0.757 + 0.437i)2-s + (0.863 + 1.49i)3-s + (−0.117 − 0.202i)4-s + (0.805 − 0.592i)5-s + 1.51i·6-s + (−0.756 + 0.654i)7-s − 1.08i·8-s + (−0.992 + 1.71i)9-s + (0.869 − 0.0961i)10-s + (−0.393 − 0.681i)11-s + (0.202 − 0.350i)12-s + 1.15·13-s + (−0.859 + 0.165i)14-s + (1.58 + 0.694i)15-s + (0.355 − 0.615i)16-s + (−0.405 − 0.703i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.343 - 0.939i$
Analytic conductor: \(3.61794\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :2),\ 0.343 - 0.939i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.04254 + 1.42831i\)
\(L(\frac12)\) \(\approx\) \(2.04254 + 1.42831i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-20.1 + 14.8i)T \)
7 \( 1 + (37.0 - 32.0i)T \)
good2 \( 1 + (-3.03 - 1.75i)T + (8 + 13.8i)T^{2} \)
3 \( 1 + (-7.77 - 13.4i)T + (-40.5 + 70.1i)T^{2} \)
11 \( 1 + (47.6 + 82.4i)T + (-7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 - 194.T + 2.85e4T^{2} \)
17 \( 1 + (117. + 203. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (87.8 + 50.7i)T + (6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (-365. - 210. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + 62.2T + 7.07e5T^{2} \)
31 \( 1 + (1.29e3 - 748. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (-161. - 93.4i)T + (9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + 395. iT - 2.82e6T^{2} \)
43 \( 1 - 3.51e3iT - 3.41e6T^{2} \)
47 \( 1 + (-607. + 1.05e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (1.89e3 - 1.09e3i)T + (3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (1.52e3 - 882. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (1.33e3 + 770. i)T + (6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (4.14e3 - 2.39e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 329.T + 2.54e7T^{2} \)
73 \( 1 + (387. + 670. i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-2.50e3 + 4.33e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 1.24e4T + 4.74e7T^{2} \)
89 \( 1 + (-4.89e3 - 2.82e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 - 5.03e3T + 8.85e7T^{2} \)
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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

−15.94183151355944850709379650782, −14.88879447240815241013791560881, −13.76505066356897025793331352679, −13.06827689442272087614378682777, −10.69762657592440411333097507835, −9.423568841232440138369182455774, −8.859478115903536219327108684808, −5.98912541915131946488858567138, −4.88604090496565933974541297091, −3.28252255664471093659620955156, 2.06264751259523291838631464869, 3.47697282048203657509341070790, 6.23410693063413153270367873766, 7.47689820828699210020775970761, 8.947816479723045231604794539411, 10.81559092240261208842331309747, 12.60325836067346005755666385668, 13.19554777158304001278734335029, 13.81602980292129063914090377502, 14.89263360882448050491983881137

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