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Properties

Label 2-35-35.18-c2-0-4
Degree $2$
Conductor $35$
Sign $0.989 + 0.146i$
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  + (0.585 + 0.156i)2-s + (4.00 − 1.07i)3-s + (−3.14 − 1.81i)4-s + (−4.51 + 2.15i)5-s + 2.51·6-s + (3.65 + 5.97i)7-s + (−3.27 − 3.27i)8-s + (7.05 − 4.07i)9-s + (−2.98 + 0.552i)10-s + (−2.32 + 4.01i)11-s + (−14.5 − 3.89i)12-s + (−6.67 − 6.67i)13-s + (1.20 + 4.07i)14-s + (−15.7 + 13.4i)15-s + (5.86 + 10.1i)16-s + (−4.93 − 18.4i)17-s + ⋯
L(s)  = 1  + (0.292 + 0.0784i)2-s + (1.33 − 0.357i)3-s + (−0.786 − 0.454i)4-s + (−0.902 + 0.430i)5-s + 0.418·6-s + (0.521 + 0.853i)7-s + (−0.408 − 0.408i)8-s + (0.784 − 0.452i)9-s + (−0.298 + 0.0552i)10-s + (−0.210 + 0.365i)11-s + (−1.21 − 0.324i)12-s + (−0.513 − 0.513i)13-s + (0.0857 + 0.290i)14-s + (−1.04 + 0.896i)15-s + (0.366 + 0.634i)16-s + (−0.290 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.989 + 0.146i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ 0.989 + 0.146i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.29473 - 0.0952701i\)
\(L(\frac12)\) \(\approx\) \(1.29473 - 0.0952701i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (4.51 - 2.15i)T \)
7 \( 1 + (-3.65 - 5.97i)T \)
good2 \( 1 + (-0.585 - 0.156i)T + (3.46 + 2i)T^{2} \)
3 \( 1 + (-4.00 + 1.07i)T + (7.79 - 4.5i)T^{2} \)
11 \( 1 + (2.32 - 4.01i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (6.67 + 6.67i)T + 169iT^{2} \)
17 \( 1 + (4.93 + 18.4i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (-22.5 + 13.0i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (2.06 - 7.71i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 39.4iT - 841T^{2} \)
31 \( 1 + (-14.1 + 24.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-33.9 - 9.10i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 20.8T + 1.68e3T^{2} \)
43 \( 1 + (14.8 + 14.8i)T + 1.84e3iT^{2} \)
47 \( 1 + (57.3 + 15.3i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (31.9 - 8.55i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-36.4 - 21.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-19.6 - 33.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (9.45 + 35.2i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 41.3T + 5.04e3T^{2} \)
73 \( 1 + (6.72 - 1.80i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (119. - 68.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (37.7 + 37.7i)T + 6.88e3iT^{2} \)
89 \( 1 + (-106. + 61.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (4.90 - 4.90i)T - 9.40e3iT^{2} \)
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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

−15.63856371241578516398605130160, −14.92147886264218814325270058738, −14.14144542609584342269077251348, −12.99946321674829362973145280426, −11.61831002023069832398638134997, −9.645958598264613854118195010365, −8.532341658092925093621316331822, −7.38475076429070677827271183695, −4.96995080343814645812913404600, −3.00130932937809305390410351220, 3.53102191252042823505587009593, 4.54715564077484876907490993728, 7.82041467013215021501110357924, 8.416446194419154850474205303365, 9.780302074898498937465228306804, 11.64996728762659879520688229247, 13.05745602064480985529531257398, 14.07056794213760300324875265267, 14.82880431769507557900409568082, 16.22904982085593712384455487927

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