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Properties

Label 2-35-5.2-c2-0-4
Degree $2$
Conductor $35$
Sign $0.919 + 0.393i$
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  + (0.109 + 0.109i)2-s + (1.67 − 1.67i)3-s − 3.97i·4-s + (−0.861 + 4.92i)5-s + 0.367·6-s + (1.87 + 1.87i)7-s + (0.875 − 0.875i)8-s + 3.40i·9-s + (−0.635 + 0.446i)10-s − 17.1·11-s + (−6.65 − 6.65i)12-s + (1.79 − 1.79i)13-s + 0.410i·14-s + (6.79 + 9.67i)15-s − 15.7·16-s + (14.9 + 14.9i)17-s + ⋯
L(s)  = 1  + (0.0548 + 0.0548i)2-s + (0.557 − 0.557i)3-s − 0.993i·4-s + (−0.172 + 0.985i)5-s + 0.0612·6-s + (0.267 + 0.267i)7-s + (0.109 − 0.109i)8-s + 0.378i·9-s + (−0.0635 + 0.0446i)10-s − 1.56·11-s + (−0.554 − 0.554i)12-s + (0.137 − 0.137i)13-s + 0.0293i·14-s + (0.453 + 0.645i)15-s − 0.981·16-s + (0.880 + 0.880i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.919 + 0.393i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ 0.919 + 0.393i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.13772 - 0.233541i\)
\(L(\frac12)\) \(\approx\) \(1.13772 - 0.233541i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.861 - 4.92i)T \)
7 \( 1 + (-1.87 - 1.87i)T \)
good2 \( 1 + (-0.109 - 0.109i)T + 4iT^{2} \)
3 \( 1 + (-1.67 + 1.67i)T - 9iT^{2} \)
11 \( 1 + 17.1T + 121T^{2} \)
13 \( 1 + (-1.79 + 1.79i)T - 169iT^{2} \)
17 \( 1 + (-14.9 - 14.9i)T + 289iT^{2} \)
19 \( 1 + 20.4iT - 361T^{2} \)
23 \( 1 + (-16.8 + 16.8i)T - 529iT^{2} \)
29 \( 1 + 2.55iT - 841T^{2} \)
31 \( 1 - 14.2T + 961T^{2} \)
37 \( 1 + (35.9 + 35.9i)T + 1.36e3iT^{2} \)
41 \( 1 + 23.4T + 1.68e3T^{2} \)
43 \( 1 + (-7.15 + 7.15i)T - 1.84e3iT^{2} \)
47 \( 1 + (6.36 + 6.36i)T + 2.20e3iT^{2} \)
53 \( 1 + (10.9 - 10.9i)T - 2.80e3iT^{2} \)
59 \( 1 + 19.4iT - 3.48e3T^{2} \)
61 \( 1 - 73.9T + 3.72e3T^{2} \)
67 \( 1 + (-86.3 - 86.3i)T + 4.48e3iT^{2} \)
71 \( 1 - 22.2T + 5.04e3T^{2} \)
73 \( 1 + (73.4 - 73.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 55.0iT - 6.24e3T^{2} \)
83 \( 1 + (52.8 - 52.8i)T - 6.88e3iT^{2} \)
89 \( 1 - 129. iT - 7.92e3T^{2} \)
97 \( 1 + (-94.8 - 94.8i)T + 9.40e3iT^{2} \)
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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

−15.82781592054645813140673024788, −14.94533413222516159623463889198, −14.02005965731327780538242995638, −12.95442036736945844821519299433, −11.01485919235026844438974888206, −10.25214781722778114703771223050, −8.317786658797831776579944514600, −7.02926929861067439315537935271, −5.34954713428449861934765447217, −2.50003234195070708078282890990, 3.37843128136223885047960577660, 4.96225976013643314528032284743, 7.66877876590620185191592992676, 8.593315251401590520642766483898, 9.943948900865475898621324855021, 11.74509745008292365907891568752, 12.80073207034368694918824036875, 13.88828493086775110691785058259, 15.50771011886671103956367169510, 16.28197842362133129018613689112

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