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Properties

Label 2-1205-5.4-c1-0-53
Degree $2$
Conductor $1205$
Sign $-0.276 - 0.960i$
Analytic cond. $9.62197$
Root an. cond. $3.10193$
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  + 2.36i·2-s + 2.05i·3-s − 3.57·4-s + (−0.618 − 2.14i)5-s − 4.84·6-s − 3.07i·7-s − 3.73i·8-s − 1.21·9-s + (5.07 − 1.46i)10-s + 0.942·11-s − 7.34i·12-s − 1.99i·13-s + 7.26·14-s + (4.41 − 1.26i)15-s + 1.65·16-s + 3.11i·17-s + ⋯
L(s)  = 1  + 1.67i·2-s + 1.18i·3-s − 1.78·4-s + (−0.276 − 0.960i)5-s − 1.97·6-s − 1.16i·7-s − 1.31i·8-s − 0.404·9-s + (1.60 − 0.462i)10-s + 0.284·11-s − 2.12i·12-s − 0.553i·13-s + 1.94·14-s + (1.13 − 0.327i)15-s + 0.413·16-s + 0.755i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.276 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $-0.276 - 0.960i$
Analytic conductor: \(9.62197\)
Root analytic conductor: \(3.10193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (724, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1205,\ (\ :1/2),\ -0.276 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.427813872\)
\(L(\frac12)\) \(\approx\) \(1.427813872\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.618 + 2.14i)T \)
241 \( 1 - T \)
good2 \( 1 - 2.36iT - 2T^{2} \)
3 \( 1 - 2.05iT - 3T^{2} \)
7 \( 1 + 3.07iT - 7T^{2} \)
11 \( 1 - 0.942T + 11T^{2} \)
13 \( 1 + 1.99iT - 13T^{2} \)
17 \( 1 - 3.11iT - 17T^{2} \)
19 \( 1 - 6.33T + 19T^{2} \)
23 \( 1 + 7.91iT - 23T^{2} \)
29 \( 1 - 1.63T + 29T^{2} \)
31 \( 1 - 6.96T + 31T^{2} \)
37 \( 1 + 0.864iT - 37T^{2} \)
41 \( 1 - 1.70T + 41T^{2} \)
43 \( 1 - 10.2iT - 43T^{2} \)
47 \( 1 + 6.49iT - 47T^{2} \)
53 \( 1 + 2.20iT - 53T^{2} \)
59 \( 1 - 7.69T + 59T^{2} \)
61 \( 1 + 3.70T + 61T^{2} \)
67 \( 1 + 7.96iT - 67T^{2} \)
71 \( 1 + 7.02T + 71T^{2} \)
73 \( 1 - 6.13iT - 73T^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 + 3.98iT - 83T^{2} \)
89 \( 1 - 11.7T + 89T^{2} \)
97 \( 1 + 17.6iT - 97T^{2} \)
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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

−9.821083860838392471376306600305, −8.960457486880178310703772070876, −8.245864847686494840753589178270, −7.60863913603161775764517599436, −6.68050120857739963103283479849, −5.72141518373833875356958301428, −4.74283144624311666932371643589, −4.44968654452477592460822691141, −3.50529323792987346893962860950, −0.793791534289983177562680923674, 1.10128658538314788779234055874, 2.18676579278323208091019999898, 2.84211790738160760610025907430, 3.75993559345854212273765070102, 5.11118146013728808284339513468, 6.21299030814697277992996290939, 7.13530882314361707728147744619, 7.85545841399893036196033422844, 9.034182880986773929699275351013, 9.586602721108183161017644833875

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