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Properties

Label 2-735-15.2-c1-0-71
Degree $2$
Conductor $735$
Sign $-0.594 - 0.803i$
Analytic cond. $5.86900$
Root an. cond. $2.42260$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.72 − 1.72i)2-s + (−0.953 − 1.44i)3-s − 3.95i·4-s + (−1.96 − 1.07i)5-s + (−4.13 − 0.849i)6-s + (−3.36 − 3.36i)8-s + (−1.18 + 2.75i)9-s + (−5.23 + 1.53i)10-s − 3.55i·11-s + (−5.71 + 3.76i)12-s + (1.28 − 1.28i)13-s + (0.323 + 3.85i)15-s − 3.71·16-s + (−2.16 + 2.16i)17-s + (2.71 + 6.79i)18-s + 0.383i·19-s + ⋯
L(s)  = 1  + (1.21 − 1.21i)2-s + (−0.550 − 0.834i)3-s − 1.97i·4-s + (−0.877 − 0.478i)5-s + (−1.68 − 0.346i)6-s + (−1.19 − 1.19i)8-s + (−0.393 + 0.919i)9-s + (−1.65 + 0.486i)10-s − 1.07i·11-s + (−1.64 + 1.08i)12-s + (0.356 − 0.356i)13-s + (0.0834 + 0.996i)15-s − 0.927·16-s + (−0.524 + 0.524i)17-s + (0.640 + 1.60i)18-s + 0.0878i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.594 - 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.594 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.594 - 0.803i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.594 - 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.738278 + 1.46489i\)
\(L(\frac12)\) \(\approx\) \(0.738278 + 1.46489i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.953 + 1.44i)T \)
5 \( 1 + (1.96 + 1.07i)T \)
7 \( 1 \)
good2 \( 1 + (-1.72 + 1.72i)T - 2iT^{2} \)
11 \( 1 + 3.55iT - 11T^{2} \)
13 \( 1 + (-1.28 + 1.28i)T - 13iT^{2} \)
17 \( 1 + (2.16 - 2.16i)T - 17iT^{2} \)
19 \( 1 - 0.383iT - 19T^{2} \)
23 \( 1 + (-1.79 - 1.79i)T + 23iT^{2} \)
29 \( 1 + 5.51T + 29T^{2} \)
31 \( 1 + 0.647T + 31T^{2} \)
37 \( 1 + (3.66 + 3.66i)T + 37iT^{2} \)
41 \( 1 + 10.1iT - 41T^{2} \)
43 \( 1 + (0.335 - 0.335i)T - 43iT^{2} \)
47 \( 1 + (-2.05 + 2.05i)T - 47iT^{2} \)
53 \( 1 + (2.22 + 2.22i)T + 53iT^{2} \)
59 \( 1 - 7.63T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 + (9.07 + 9.07i)T + 67iT^{2} \)
71 \( 1 - 3.06iT - 71T^{2} \)
73 \( 1 + (2.32 - 2.32i)T - 73iT^{2} \)
79 \( 1 + 3.70iT - 79T^{2} \)
83 \( 1 + (-0.973 - 0.973i)T + 83iT^{2} \)
89 \( 1 - 3.03T + 89T^{2} \)
97 \( 1 + (10.3 + 10.3i)T + 97iT^{2} \)
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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

−10.49435408074911673998624020475, −8.964327780302364044959222371726, −8.105759734056667419190246855502, −7.00833982204114269179451755129, −5.78750768568406646651480264104, −5.29140425676063947531621760124, −4.07878481919588261425698195968, −3.27457209672981572796244652809, −1.88654844112421808353125234597, −0.61264737917296682404893760825, 3.00654952656391127290853704274, 4.07113608924823373347449177761, 4.56713233147347758789877653631, 5.46121951462680484041084963194, 6.60491561586901417227080133279, 7.01727992752471067386145559056, 8.036533542286988173029154042131, 9.074621795418814681765286942457, 10.16120248752031396735590453683, 11.21034158155187693045188875951

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