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Properties

Label 2-65-65.12-c2-0-5
Degree $2$
Conductor $65$
Sign $0.882 - 0.470i$
Analytic cond. $1.77112$
Root an. cond. $1.33083$
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  + (0.474 − 0.474i)2-s + (−0.839 + 0.839i)3-s + 3.55i·4-s + (4.99 − 0.121i)5-s + 0.796i·6-s + (1.35 − 1.35i)7-s + (3.58 + 3.58i)8-s + 7.58i·9-s + (2.31 − 2.42i)10-s − 7.57i·11-s + (−2.98 − 2.98i)12-s + (1.13 − 12.9i)13-s − 1.28i·14-s + (−4.09 + 4.29i)15-s − 10.8·16-s + (−4.65 − 4.65i)17-s + ⋯
L(s)  = 1  + (0.237 − 0.237i)2-s + (−0.279 + 0.279i)3-s + 0.887i·4-s + (0.999 − 0.0243i)5-s + 0.132i·6-s + (0.193 − 0.193i)7-s + (0.447 + 0.447i)8-s + 0.843i·9-s + (0.231 − 0.242i)10-s − 0.689i·11-s + (−0.248 − 0.248i)12-s + (0.0873 − 0.996i)13-s − 0.0916i·14-s + (−0.273 + 0.286i)15-s − 0.675·16-s + (−0.273 − 0.273i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.882 - 0.470i$
Analytic conductor: \(1.77112\)
Root analytic conductor: \(1.33083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1),\ 0.882 - 0.470i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.37005 + 0.342840i\)
\(L(\frac12)\) \(\approx\) \(1.37005 + 0.342840i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-4.99 + 0.121i)T \)
13 \( 1 + (-1.13 + 12.9i)T \)
good2 \( 1 + (-0.474 + 0.474i)T - 4iT^{2} \)
3 \( 1 + (0.839 - 0.839i)T - 9iT^{2} \)
7 \( 1 + (-1.35 + 1.35i)T - 49iT^{2} \)
11 \( 1 + 7.57iT - 121T^{2} \)
17 \( 1 + (4.65 + 4.65i)T + 289iT^{2} \)
19 \( 1 + 25.1T + 361T^{2} \)
23 \( 1 + (-8.91 + 8.91i)T - 529iT^{2} \)
29 \( 1 - 16.0iT - 841T^{2} \)
31 \( 1 + 33.0iT - 961T^{2} \)
37 \( 1 + (-46.0 + 46.0i)T - 1.36e3iT^{2} \)
41 \( 1 - 32.9iT - 1.68e3T^{2} \)
43 \( 1 + (-15.7 + 15.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (-8.32 + 8.32i)T - 2.20e3iT^{2} \)
53 \( 1 + (44.5 - 44.5i)T - 2.80e3iT^{2} \)
59 \( 1 + 59.1T + 3.48e3T^{2} \)
61 \( 1 - 28.3T + 3.72e3T^{2} \)
67 \( 1 + (72.9 - 72.9i)T - 4.48e3iT^{2} \)
71 \( 1 - 127. iT - 5.04e3T^{2} \)
73 \( 1 + (28.4 + 28.4i)T + 5.32e3iT^{2} \)
79 \( 1 + 96.0iT - 6.24e3T^{2} \)
83 \( 1 + (47.5 + 47.5i)T + 6.88e3iT^{2} \)
89 \( 1 - 88.2T + 7.92e3T^{2} \)
97 \( 1 + (58.1 - 58.1i)T - 9.40e3iT^{2} \)
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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

−14.49909562280973158800551717223, −13.34651510958768992053878916947, −12.82410437880032864221437386077, −11.17638850752053697920160784438, −10.50875354243744020638744045439, −8.886150312149320476269600776619, −7.70104677036304223000728207834, −5.93352206986766150126741463079, −4.52717290795427255370730163926, −2.59007701909406847888867734762, 1.75425157681615865792379919871, 4.65234755088587437251625422114, 6.10836765227275356880928802440, 6.78238961215267400380753471162, 9.006834649171545823539670027018, 9.915763920942032098609332051062, 11.12592611738180157054091320225, 12.51467258502695905768858510351, 13.57404846820714025897571850884, 14.62330889912293965332686402292

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